I understand the fight to get them to do what they need to do. My oldest is an Aspie also. We didn't have Tourette's. I think everyone with a form of autism has sensory integration disorder/processing disorder. It is hard to live with them when they are raging.
I guess my kids got off luckier than some, I have my own severe sensory issues, especially regarding textures and foods/smells. I was a vastly pickier eater than any of my kids, lol.
I always kept the thought that my job as a parent was not to make my kid happy. It was to make my kid into a contributing member of society (Words straight out of my mother's mouth, lol - I MUST be old now! lol!). I fought him to learn many things, like how to greet a stranger, even if he didn't like something about them he still had to be able to greet them properly. He had to learn basic manners, whether either of us liked it or not. He had to go to school and get an education at least through high school. Like it? Not like it? Not my problem. I was active in his schools and fought for him when it was needed, but I fought for his teachers/classmates also.
One thing we found was that letting him avoid everything that bugged him just restricted his world. If we let him avoid tomatoes, there would be something else he would insist on avoiding. Or three something elses. He had to learn how to do chores because life is life. It was often a knock down screaming fight because he could not have his tv time until/unless he did the chores.
I often told him that if he wanted to scream, to go scream in his room. My own mother told me that if I wanted to scream, to go scream in the hall. Of course it made me angry. The whole point of screaming is to get attention and make your problem into someone else's problem. It made my son angry also.
If your son is 16, he is old enough to pay for his own things through work he does to earn money. If your son wants sole control over the cat, he needs to be earning money to pay for the cat's food, litter, and other bills. Otherwise, the cat belongs to the entire family and if his sister wants to pet it, she can.
if he throws screaming fits over the internet, turn it off. I know he will have a fit no one wants to deal with. Leave it off until such time as he learns that EVERY fit over the internet will result in days/a week/weeks/a month with no internet access at home. It will be super hard at first. Do special things with the rest of the family to make up for havign to deal with his fits. Not giving consequences because you are too worn out to deal with his current fit just teaches him to be badly behaved more often. been there done that. It truly ends up badly.
I operated with a very specific rule - if something was important enough for my child to lie, cheat or hurt someone over, that item was WAY too important and unless it was air, food, shelter, basic clothing, that thing was going to be GONE. I took tv/internet away for weeks/months at a time when needed. I hated it, but it was what we needed to do. I felt that my discomfort was part of parenting. My husband once joked that it was like watching me go through labor again to endure no internet in the house AND no tv while the kids were awake. I agreed, but we did it anyway.
How much are you giving in to his germaphobia/shrink wrapped food stuff? The more you give in, the more amped up it will get. If he wants shrink wrapped food, he can get a J.O.B. and buy his own. Keep healthy food on hand, make appropriate meals available, and then ignore cries of "I'm hungry and there is no shrink wrapped food". He is clearly healthy, and if/when he gets hungry enough he will eat. Or he will decide he doesn't want to spend his own hard earned money on that and would rather spend it on something else. At 16 you have that option. He can mow lawns, shovel sidewalks, help with gardening, etc... for neighbors, or he can go and get an actual job at a store or restaurant. Work is healthy for them. It gets them into the real world.
Maybe consider making access to the internet something that he can ONLY have after he has been to school and gotten his planner signed, or some other report from his teachers. With Asperger's and Tourette's, he probably has an IEP and signing a planner can be put into that. Remember, they MUST meet to revise it if you ask them to. The school does not have a choice if you call an IEP meeting. They MUST meet.
Or make access to special shrink wrapped food the reward for going to school. Whatever motivates him.
Yes, it seems like I say to take the internet away and to use it as his carrot at the same time. If you tie it to one behavior, find soemthing else to tie to the other one. Or make tv separate from the internet and tie each of them to a different thing.
What is being done for his sensory issues? Have you seen an occupational therapist lately for help/advice? What sensory things did he stop doing that might help if you start them again? He is a little old for the sensory things that little kids do. I will say that the book "The out of Sync Child Has Fun" has a lot of activities for various sensory needs. They may be aimed at a younger age, but with creativity you could adapt them to an older child. At one point we would have every kid from our apartment complex in our tiny backyard if we got that book out. They would range in age from toddlers to high school. I swear somehow they just all KNEW if we started doing something from the book and then we would be almost mobbed with kids wanting to play. Hubby and I thoroughly enjoyed this and came to expect it. The activities were fun for us, my kids with sensory issues (all 3 of mine have them) AND they were fun for the "normal" kids too.
When your son was little, did you do brushing therapy with him? It involves using a dry surgical scrub brush over the body in a specific pattern, followed by gentle joint compressions. This is sometimes called Wilbarger Therapy, I believe. It is one of the best therapies for sensory integration/processing disorder. It helps the brain learn to use sensory input in a better way. You don't try to learn this on your own, it MUST be taught by an occupational therapist. If you do it wrong, you can create HUGE problems. Done correctly, the change is fast and incredible. If I had not seen the changes myself, I would not EVER have believed. I especially would not believe that this can be done under or over clothing, with zero medication and no high tech anything needed!! At the time I learned about this, I was told it would not work on anyone over 12-14. I thought that was odd. Our brains are not static, they change and modify themselves often as we learn and grow. I was taught to do this on my youngest. I did it with him with incredible results. I did it on myself and was surprised at how textures taht would have been reupulsive and horrible were not good in any way, but did not make me shudder or gag. I then tried doing it the way I was taught, on my older 2 children. It made a substantial difference to each of them, also. I don't know if they still say it won't work if a child is over a certain age (it was 15 yrs ago), but it might be worth finding out.
One dietary thing we found was incredibly helpful. We INSISTED that our kids had to have regular protein intake. Without protein every few hours, they got grumpy and often quite difficult or horrible to be around. If they had sugar and no protein, living with the boys was miserable. My daughter was never as bad, but that is just J. The boys could not begin to cope with the world around them if they did not have regular protein. I would buy protein bars in bulk and keep them locked up. Otherwise the kids all ate them like candy on the last Halloween they would ever have. I kept protein bars in my purse, the car, they got a few to keep in their lockers/bookbags, etc... If we had to run errands after school, they got protein bars, popcorn chicken or burgers, or some other high protein snack. If they didn't have this, they could not behave well in public. I generally looked for bars with roughly 40% carb, 30%fat and 30% protein, or were even higher in protein. Paying attention to their protein intake made an enormous impact on their moods and ability to cope with the world.
My oldest is now working full time at a grocery store. He is grateful we taught him to stay on top of his protein intake because if he eats junk, he does not feel good physically OR emotionally. He isn't a super healthy eater, but if he is going to eat junk, he will make sure he has protein first. FYI, my son was incredibly violent (oldest, Wiz). He spend several months in a psychiatric hospital as a preteen for trying to kill his sister (to the point we had to take her to the doctor and call CPS). We had to have him live with my parents a couple of years later because he was getting more and more violent. We didn't just sic him on his grandparents. They were only a couple of miles away, and my father had just retired from teaching jr high for decades. Gpa and Wiz are incredibly alike, and if anyone could out stubborn my son, it was my father. My mother managed to get him to conform to social rules in some clever ways. Somehow, it all worked and he is a self-sufficient, self-supporting, contributing member of society. He also has repaired his relationships with each of us. It took into his early 20's to get there though.
As for social rules and not coping well with injustices that are perceived, been there done that. For some of the rules, I just told my son that X was the rule, that I was aware that X made zero sense, but we were going to have to follow it anyway. Other people don't understand when we don't follow X, and that means that if you want to get your way, you give in on rules you find stupid, like X. At one point X was shaking hands when you meet someone new. It was also to look at people who are speaking to you. X was a LOT of different things over the years! My mother became his guide to social norms in his teens. She sat him down and explained that he may NEVER understand the social rules, but they are there and it is embarrassing when you don't follow them. She asked if my son knew why Gpa always asked if this shirt matched those pants? Wiz told her that Gpa had to ask because he is color blind. Then my mom told my son that having Asperger's is like being color blind for social rules. Gpa had to accept that red and green exist, are different, and can be very important. Wiz had to accept that social rules are there, are annoying sometimes, make people feel more comfortable, and often help you get what you want. Ever since then, Wiz has asked when social stuff confuses him or doesn't make any sense. I don't know if this approach will work with your son, but it is one of the best explanations I have come across for explaining the whole "why do they exist" of social rules.
If you are in the US, I STRONGLY recommend that you take one very firm, non-negotiable stance with your kids. If they live in your home, are on your insurance, and/or supported by you, you have access to their medical records. It really is helpful long run, and it can save huge fights. Sooner or later some office worker will explain that HIPAA gives them the right to keep you from seeing their medical records. I don't advocate that you violate their trust, but if your child gets very ill, you can end up with no way to help them. My kids were all told starting as preteens that if we supported them, our names go on the medical papers. It is a just in case thing, not a monitor everything at every dr visit thing.
I hope something I wrote here helps! Use what is helpful, don't worry about the rest.