What to do about breaker that triggers main panel, and breaker that's stuck?
QO's are quality breakers, so your "off breaker" is probably a tripped breaker because the circuit it's feeding has a big short.  When you reset it, the short was strong enough to also trip the main breaker.  (i.e. trip the main first).  That sometimes happens.  The repair here is to hunt down and fix the short.  It's either amateurish or failed wiring at a junction box, or a nail driven through the Romex. 

Your breaker that won't turn off is a very serious problem and that breaker should be pulled immediately.  It is an old-school "double-stuff" breaker.  The new style QO "double-stuffs" look different, the switches are abreast of each other.   That is fine.