1) If a hero inflicts damage on a minion and escapes leaving the minion with another hero who was on the same space, does the minion heal before the other hero has to fight it?
The rules from page 17 simply says the other hero will now have to fight it, it doesn't specify whether or not the minion is fully healed.
Intuitively, it feels like it should be one continuous fight, with no healing, but that's not the way it works.
Here's a link to Jason Hill's answer. He says the minion heals as soon as the first hero escapes. It's not specifically stated, but the second hero is apparently considered a different fight, rather than a continuation of the first fight.
2) If a rats nest is at the town hall does it automatically advance the shadow track when it spawns a rat or only when it spawns a second rat?
Per the Rat's Nest rules on the Werewolf's Minion card, you place a rat on the first available space that does not already have a rat or rat's nest. If you can't, then move the shadow track. So, I'd say the first rat (and each one after) causes the Shadow track to go down.