1) No. The 240 should be on its own breaker with no circuits 110
tapping off
of it
2) a)Yes,if you have 2 seperate breakers for a 240 circuit, one will
trip if
overloaded.
b) If I am understanding you correctly, the ganged breaker handle

does not move when tripped. I would say you have a defective
breaker.
3) Yes breakers do wear out over time. Federal Pacific breakers are
known to be defective. If you have this panel,change it.
4) What is the rating of the subpanel in the garage? If its at least
40A you should be ok. Just re-wire the sub-panel so the 240
plug is on its own breaker and the 110 circuits on there own breakers.