When I want to release a pet, it says that if it's over a certain number of days old, releasing it will increase my chances of finding nests. Is that bonus permanent? If not, how long does it last? How many pets have to be released to see a significant increase?

the released pet must have existed for more than 42 days.
The bonus lasts until you encounter a nest in either the forest or caverns of the LG (I do not beleive it affects the temple?). Once you find one, the bonus resets.

I am unsure on the numbers of how many make a difference, so I can't answer that last one

ah, thanks!

I can build on what Scath said. I have noticed that eggs can be triggered from anywhere as low as 50 pets released but when i used to release for eggs i would pause every 100-200 and would trigger an egg in < 5 mins most times

What is the LG and what is the forests?

the LG is the Lost Grove, which is the area that becomes accessible after you complete Astryl's Tranquility. The Forest is the first section of the LG.