Son's coach leaving the program...

Hard to know what goes on in another man's head, but if he's been good for his current program, there's no reason to think he'd slack off now. You are in a better position to look at the character of the man for that answer than anyone here.

1.) I would expect 100% from him in his current job. If it looks like he's giving anything less, I would call him on it.
2.) I'm not clear, but I was guessing this is HS, not college. Is recruiting allowed? If you're talking college, anyone he recruits will be looking at the choice between an established program and a brand new one. I would think the better players would go with an established program.
3.) Who's paying him? That's who he should work for. If he wants to focus on establishing his new position, he should step down to do so. His primary focus is on what's in front of him either today or next year. That should determine his position.
4.) Again, it depends on the character of the man. If he has any at all, he won't have that attitude.
5.) He might. But do those kids want to change schools? (Assuming High School here), We had a new program start around here a few years ago. The only kids who transferred were the ones that were in the immediate area, or the ones who weren't seeing playing time. That latter group were hoping to get some at the new school, even knowing it would be a lesser program to begin with. They just wanted to play. Most of the juniors stayed at the old school for their senior year.
6.) This year, you have a team and the other school doesn't. I would expect the same out of the team as if the other school was not in the picture. Maybe they will try harder to send him out on a winning note. Next year, they're gonna want to take it to him. Parents/Boosters should keep on with what they are doing. If some are, indeed, planning to move their child, they may become less involved. Expect that and understand it from their point of view.

Tell the kids to relax and have a good year.

Update:
Coach resigned last week after a less than stellar 1-16 start. Assistant was named interim head coach.
Team went into their first conference series this past weekend and scored 48 runs in 4 games and swept the series. I will add that my son crushed a monster 2-run pinch hit bomb in his only AB.
The team seems to have a fresh outlook on the season and we will see how things go from here.

Sooooooooooo....sometimes it IS the coach.