All of your list apart from 4 of them, I would describe as my dd as having as a toddler, she was diagnosed with high functioning autism at 11.
Don't freak out too much about the food situation. Even now my dd will only eat a very limited number of foods and only 2 vegetables and 2 fruits.I made sure she had vitamins every day.A lot of it is to do with texture. Write a list of foods your child likes and just vary it slightly with other foods they like on different days. With my dd I would ask if she would try a bite or 2 of a different food each week - if she didn't I didn't stress over it. My dd sticks to water or one type of squash and that's it which is fine by me.
Hypersensitivity to noise is a problem. My dd couldn't cope with hand-dryers either so we didn't use them. She would wash her hands then we would dry them outside of the toilet area with a couple of paper towels I kept in my bag.
Soft to touch clothes are important and a comfort for sensory processing. It took me years to realise why my dd wasn't wearing a lot of the clothes I bought. I eventually realised she always chose the soft chenille type fabrics with no seams.to this day she will not wear jeans. To her they are stiff and uncomfortable and annoy her skin.The softer the clothes the better.
Toothbrushing- get a fun toothbrush and a timer. Kids with ASD like routine. Tell him it's a game to see how well he can clean his teeth in 2 minutes.My dd liked the strawberry sweeter tasting kids toothpaste never the mint.
The hairbrushing/haircuts thing can be a sensory issue. There are some hairdressers who accommodate kids with ASD. If not have a hairdresser come to you and give him a figit toy with lots of praise when his hair is being cut. You may need to change whatever your using to brush the hair. My dd prefers wide tooth tangle tamer brush. Could you cut his hair? I cut my dd's hair she did go to the hair dressers as a toddler but for years now she wants me to cut her hair.
Not coping when things change can be an issue. It's always good to explain beforehand what's happening next but also say what may change lady minute but the change will be ok and he will be safe. Maybe put a planner on his wall so he can see what is done on different days.
The staying at home preference is common. Too many people in social situations can be overwhelming to a child with ASD. My dd now still prefers being at home. If you write when you are going out in the planner this may help.
My dd still has obsessions with certain TV programs. This is quite common. What helped though was that she only watche them at certain times then did something else.
The cold bath water is probably a sensory issue on the skin.unless it's going to cause a health problem I wouldn't worry about it. My dd's showers are freezing in comparison to mine.As long as they get clean it doesn't matter.
Meltdowns can be common in toddlers anyway but overwhelm can cause meltdowns in kids with ASD. Keep explaining it's not ok to hit, stay with him in a quiet area until he calms down maybe use a distraction figit toy to de-escalate.