Cheborey JochieSantiago On the GitHub page, if you scroll down, there's a Q&A chapter. Please check the section: "No Pokémon are appearing for me, what's wrong?" for some tips that may help. The Q&A also addresses your other questions. In short: Pokémon stay, and glancing at your lock screen is fine.
The timing is randomized, but as a general rule, you should find around 2 Pokémon per hour of consecutive screen-off time. The soonest they can appear is after 15 minutes. (The only exception is your starter Pokémon.)
