I think that you are:
I'm certainly not the best in the room
but I do what I can and results are good.
If you come in second you are only second best (not the best) - but in a wide field of entrants coming second is still a good outcome.
I'm never around for too long in the end,
wouldn't stop for a minute to breathe, if I could.
A 'second' is a very short period of time, and certainly much shorter than a minute.
My color is silver, my place to the left.
Second place in many sporting events is rewarded with a silver medal. On the winner's podium the silver medallist traditionally stands to the left of the winner (as seen face-on by a spectator).
Rulers call me their vice, I'm who they rely on.
'Second-in-command' is another word for 'vice', as in 'vice-President' - the person closest to the leader and working to support them in their objectives.
I'm fleeting and small, hard to track, quick to leave:
If you count to ten I am already gone.
As in the second (!) rhyming couplet, a second is a small unit of time which passes quickly. Very few people can count out loud to ten without a second elapsing!