1: The safty is not on and the trigger guard is disengaged... which probably means the hammer is not in the rearward "cocked" position... this is probably how the trigger gaurd got bent and why it was really hard to take it off the first time... when the hammer is not cocked and you try to take the trigger guard off... you are cocking the hammer like a lever action rifle... which would be pretty hard with out any leverage on that short trigger guard and you probably put alot of weight / pressure on it... thus bending the trigger guard.
2: It appears the trigger guard is slightly bent... thats why its not fully engaging the latch and that is why the trigger guard is popping open when you fire.
3: You will need to slightly bend the trigger guard back... do this very carefully. Get a couple of thick socks to cover the metal and use some pliers... frequently checking to see if it snaps together and stays put. When it snaps closed... and is some what difficult to take off it is fixed.
4: The hammer should be in the cocked position... and the safty on... to remove the trigger group. Now... go to the local hardware store and get a peice of thick copper wire... thick enough wire to where you cant bend it (copper wire about as thick as a pencil or so). It needs to be copper so it wont chip or bend or damage the metal or its finish... use that to stick in the hole of the trigger guard (found in the rear end of the trigger guard) and use it to slightly bend back the trigger guard untill it pops open. That is how to properly break down the trigger guard out of the action...
Any other questions about the Mini-14 just ask... i have been shooting them for 18 years.
Bye the way... if you like to shoot your Mini-14 alot... here is a good tactical sling for it... make sure to select the Mini-14 option on the right side of the page when ordering... i have one of these for both of my Mini-14's... they work great and are well made.