Within Pokémon Go, when you throw a Pokéball at a Pokémon, you have the potential to get a Nice, Great, or Excellent throw. If you throw an excellent Pokéball on a challenging Pokémon (when their circle is orange or red), will this increase your chances of capturing the Pokémon?
Getting a "Nice", "Great", or "Excellent" throw is possible by getting the Pokeball to land inside the shrinking colored circle. This gets you bonus experience if you catch it with that throw. The size of the circle determines which one you get. (Approximately, full circle-66%=Nice, 66%-33%=Great, 33%-smallest circle=Excellent.)
So getting those special throws won't boost your catch chance directly but having the circle as small as possible will.
The official FAQ says:
You have the greatest chance of capturing the Pokémon while the colored ring is at its smallest diameter.
After looking at many sources online and the research that others have done, making an "Nice", "Great", or "Excellent" throw does increase your chances of catching the Pokémon.
Looks like the throws bonuses are:
Nice Throw - 1.15X bonus
Great Throw - 1.5X bonus
Excellent Throw - 1.85X bonus
Here is a catch rate calculator so you can play with the odds yourself.
I don't believe so. I have no concrete proof of this, and while a lot of people say this is true, I'm not convinced. What I've read is that the smaller the inner circle the higher the catch rate, but in my experience it has no meaning other than the opportunity to get extra xp from the nice/great/excellent throw. I can't say for sure, as i get great/nice a lot more often than excellent, but i have had Pokemon escape and excellent throw, even with low CP and a green inner circle. That being said I still try to throw it when the circle is the smallest, maybe I'm just superstitious.