1) How do you all feel about this treatment/situation?
Hopefully it is just a beginning of the season type of situation. I am wondering what the coach would plan on doing in case of injury/sickness. I bet he hasn't even thought about that. At all levels of baseball there are at least 2 catchers.
2) Should he catch both games of double headers plus all this practice?
Do they have doubleheaders all the time? I think catching a DH once in a while is ok, especially if it is a 7 - 5 inning DH. 7-7, or 9-7 is too much. A lot depends on where your son is at growth wise. If he is pretty much grown, then I think that it is acceptable over a short period of time. When he gets to the college level it will be a lot of work for a much longer than HS schedule.
3) How should I approach the coach?
Your son should approach the coach and ask about getting some help in the bullpen and preparing a backup.
4) What should I say to the coach?
If the coach does not respond to the player then you may choose to get involved.
I will say for my son his Freshman year he was the JV catcher (2-3 games per week) AND the bullpen catcher for the Varsity doing all game bullpens and practice bullpens, in addition to a few Varsity starts and limited DH situations. He has slowed running wise but his legs are really powerful and well suited for the position, but I think that is inevitable for a catcher.