1. Should these two breakers be ganged together?

Yes. I'm reasonably sure the electrical code requires it. In
any case, it's the safe thing to do.

2. If one trips this effectively shuts down the heater
doesn't it? That
is, the heater will no longer provide heat.

Yes, tripping one breaker will stop the heat from comign
out.

3. Is there still a problem since the 2nd breaker hasn't
tripped? I mean
other than a possible danger to someone futzing around in
the heater's
innards.

The wires are still electrically charged, so this is a
safety problem for someone futzing around in the heater's
innards, trying to figure out why it's not heating. Double
pole breakers aren't that expensive. Since you're asking
these questions, I'm guessing you don't feel comfortable
replacing your own breaker. Call a friend who is, or an
electrician.