Your lovely little ds sounds so much like my ds (8) at that age. I really understand how physically and emotionally draining it can be when our children won't just 'join in and get on with it'. My parents used to drive me mad by constantly asking me "what's the matter with him?"
Some children are just very sensitive and need a whole heap more patience and really can't just be chucked in at the deep end. They do get there though. It often just takes time, and a different approach.
walking We never use the 'W' word. Telling ds we are going for a walk is guaranteed to make him go limp and start whining. We go, exploring, on an adventure, for a trek, a hike, a race, a run etc. We also quite often don't tell him where we are going, just what. So, out for a picnic, to find minibeasts, to a café, to take photos, or nothing at all. I've learned that sometimes keeping information to an absolute minimum can be the best route forward. Maybe take the trains on a ride through the woods?
bathtime. What worked for us was just picking ds up and putting him in. No discussion about it. A really fun bathtime with the action men (could he bath some of his trains?). 1 minute warning about getting out, followed by pulling the plug if he didn't comply, again no discussion. Lots of warm towels around for when he did stop shouting and finally got himself out.
Hairwashing give him a towel to hold against his face and get him to tip his head backwards, wash his hair quickly then rinse quickly with a cup, not the showerhead. Rub his hair dry while still in the bath. Ds hated those trickles of water that run down your face/neck after you've washed it.
swimming The swimming pool can be a massively scary place. My dd had a terrible fear of the drains in swimming pools that at one point (before she could articulate it properly) it also included toilets and baths. It took 18 months to go from her not being able to leave the changing room to returning to swimming lessons (ages 4 to 6.5). I think you've just got to be massively patient and encourage him to go a little bit further each time. Again, could the trains go swimming? We took some barbies in with dd, which got her from the edge of the pool to knee deep. I also found a pool with a graduated shallow end, which really helped as there was no 'getting in'.
bike / scooter / music class Don't push him. He'll do stuff like this when he's ready.
won't wear shorts Get some long linen or thin cotton pants. These things just aren't worth the battle. My friends little boy would only wear long sleeved pyjama tops for about a year. He's 7 now and wouldn't be seen dead in a long sleeved pyjama top. 
parties Take him, but explain to the mum when accepting the invitation that he finds parties quite stressful and that you might not stay. Ignore any 
responses. ds still hates loud chaotic situations, so I sometimes turn down an invitation if I think It'll be too much for him. He mostly loves parties now though.
food festivals/craft markets Again, noise, too many people, over-stimulation. Don't take him, or take him for a very short time. I find if ds has a book to read we get much longer out of these events now, as reading gives him some 'zone out' time. He wouldn't have coped with this kind of thing at all at 4, or it would have been a full time job for me to keep him on an even keel, so not worth it.
Good luck op.