Not at all. The leagues matter only when you make it to the Legends, or at least that's what it is believed. Besides they don't show anything except for a fact, that once you spent enough time playing in one week to make more virtual points than a number of other people.
What does Player's League do?

It's just for fun. You can even exit from league at any stage. It really doesn't help you in any way!
If I exit the league, do I start from Wood all over again or do I keep my original league?

not really, the only league which kinda matters is legends as there you can win money but its extremely hard considering you must be in the top 3 to win money which again is difficult

It's just for fun. You can even exit from league at any stage. It really doesn't help you in any way!
If I exit the league, do I start from Wood all over again or do I keep my original league?
If exit and rejoin, you'll be at the same level you are when you left. Unless something changes in the future.
I recently reached Crystal in Player's League.
Does it affect my games and levels in any way?