Does the timer above the gym have to reach zero for the battle to begin or can you start it as soon as you are in range of the gym?
There are two types of timers. One that appears above an egg, which the amount of time you have until you're allowed to enter the Raid, and the other which is above a Pokémon, which shows how long you have to defeat the raid before it disappears. So you have to wait for the first timer to hit zero and for the egg to hatch, but once it does, you're free to battle once you're in range until the second timer runs out.
Is it possible for me to get in the way of other (better) players or can I just join any gym battle?
Given the context, I assume you mean Raid battle, not gym battle. Anyways, a Raid can have up to 20 players participating at a time. And so long as you're not in a group with more than 20 people, you're presence will only help the group, not hold them back in any way. You may not help that much, but you won't make matters worse.
Do I or should I try to find the other players and coordinate with them?
Depends on what your area is like. I play in two different types of areas. At home, I live in a rural community where it's impossible to do high level Raids without first planning ahead with other players via a Face Book group. But at college, there's so many people playing that if I show up at the beginning of a Raid, odds are there will be people there, therefore I have no need for coordination with other players.
Since you mentioned you want to Darkrai, it's worth noting that since Darkrai is special limited-time event, if your community does have a group that consistently plays, they will be participating. I suggest you go to most well-trafficked gym in your area, such as at a park or shopping center, at the beginning of the event, and see who shows up. You can talk to them about how people in your community typically play.
When I did the Mewtwo gym raid all the pokemon I selected for the battle fainted but I was able to start over and did eventually get him. Can we basically keep trying until we are out of pokemon?
You can keep on doing this until the battle's timer runs out, which is about two minutes. Once that timer runs out, if you didn't beat the Raid, you're welcome to heal up and try again without needing to spend another Raid Pass, but you'll have to start the battle over again with the Raid Boss fully healed.