Hello op. My daughter is 6 now so I don't know if any guidance has changed since I had her but I will tell you the best of what I know!
1 - sleeping. Some babies need more sleep then others. My dd was a terror (even now she gets far less sleep than most of her peers). The best you can do is try to implement a bed time routine, keep things calm and relaxed, feed him, put him down. You can't stop him from waking up, but you can try to make sure everything is calm and quiet so he doesn't get too awake. Some people do enforce stricter routines and do things like "cry it out" - you can Google to read more. Tbh I was exhausted and just ended up co-sleeping with my DD as she was bf, screamed every time I tried to put her in her cot, and I got more sleep that way.
Some people are very against co-sleeping. There's also the "shush pat" method. It's really a personal choice how you manage it and you have to do what feels right for you and your ds. It won't go on forever. Sorry I'm not sure what you mean by playgrounds etc where he can sleep. You can get those baby chair things, my daughter sometimes fell asleep in hers.
2 - food. Let him guide. If he only wants a couple of spoonfuls, that's fine. Read about baby lead weaning too - it's basically offering him things he can pick up - like chopped veg/fruit, toast, chicken etc. No bones of course. Don't worry too much about how much he eats, naturally as he gets bigger he will eat more. "Food is for fun until 1" so it's just getting him getting used to textures and tastes, not so important nutritionally as he gets what he needs from milk.
3 - as far as I understand, water isn't necessary for breastfed babies. If he is on formula than water is more important. Breastfed is not, but I was giving my DD water at that age to try and cut down breastfeeding a bit (for my benefit). Probably no more than one cup a day - my memory is a bit fuzzy on that, but I don't think you need to worry if he doesn't like it yet. Just try giving it to him every so often and see if he'll drink any.
4 - I'm assuming things are trickier with weigh ins because of the pandemic. Unless you have any obvious concerns like he doesn't seem to be gaining weight, looks dehydrated, isn't producing wet and dirty nappies, then I would say try not to worry about it. If he stops producing wet or dirty nappies, or shows any signs of dehydration then you can speak to them as a matter of urgency. It sounds like you're quite anxious, it is a difficult time to have a baby what with every thing going on and I feel for you, but try your hardest to relax. It helped me to remember that women have been having babies forever, for far longer than we have had weekly weigh ins and all the other modern interventions we have now. As long as you're keeping him fed, clean and loved you can't really go wrong. Any major concerns you can get him taken to a GP or a&e or something, but I'm sure that won't happen.
Sorry for my long post but I hope there was something helpful for you there. Always ask on here if you have any specific worries.