what could be wrong with the car when you have ...

If you have a hard time putting it in reverse, that could be a transmission problem.  If you have to shake the key in the ignition, you could have a bad ignition switch.

what could be wrong with the car when you have a hard time putting it reverse and have to keep pushing on brakes and shaking the key in the ignition...

this is a strange question? heres why... a hard time in reverse does the car stall when you put it in reverse? if so the jerk of the car can cause the safty/neutral switch on transmission, to break contact and stall. if while car is running and your shifter is loose and not going into gear properly and you wiggle shifter and it finds it way in position. now shaking key has nothing to do with reverse. the hitting of the breaks part may cause the safty/neutral switch to move slighty in position and help?? it is unclear if its an electrical problem or mechanical problem your having from your question?? you never said the car stalls or is just hard to put in reverse while car is running??

I think there have some problem on your ignition switch