1) If you don't have a Pokémon on the Gym, you have to be in range of the Gym to feed Pokémon berries; if you can't spin the stop, you can't feed it berries. (edit; forgot to add this part. my bad) If you do have a Pokémon in the Gym, you have to go to the gym through your Pokémon itself in your Pokémon Bag (tapping on the Pokémon, then tapping on Go To Gym) in order to feed it from a distance.
2) There is a Speed limit - I believe it is 80km/hr - for catching. If you're going too fast you won't catch anything.
3) Not sure - I'd have to see a screenshot to know what you're talking about.
4) They aren't supposed to be event exclusive. Once they're introduced, they should be in the game permanently. But I can't confirm that because I never (censored) catch (censored) Shiny's outside of (censored) Community Day or one Makuhita. (Except for Swablu, but I don't remember if Swablu had an event associated with its Shiny form.)
5) Likely because Niantic has many raids at Sponsored Gyms, and many of those locations close in the evenings. Niantic doesn't want people congregating at a sponsored location at 10pm if the location isn't currently open for business because it's a security risk for the location and the foot-traffic to the location is useless to Niantic and the Location, so Niantic doesn't get paid for it. This is also a big reason that Spoofing is considered a problem by Niantic; they get paid by some Locations for every activity on the Gym because it's assumed that the activity on the Gym means the person is coming into the Location - entering the Starbucks, for instance. And Niantic probably doesn't want to be responsible for problems if, say, 20 people gathered at 11pm in a public park to do a Ttar raid and the police think it's some sort of gang activity because why the heck are 20 people in a park at 11pm so let's just harass and arrest everyone then figure it out down at the station. (yes, hyperbole, I know.)