Good grief your starting off the New Year on the wrong foot.
Lets start by learning more about your vehicle, How many miles do you have on it?
Do you have a check engine or other service light on?
It sounds like you are describing a couple different problems.
The starting problem and the lock issue may be related to and part of the security system malfunctioning.
I would suggest searching any and all related TSBs technical service bulletins found in a online subscription to a factory service manual like Alldata diy or Mitchel 1 diy
If you do not find any related TSBs we can move on to diagnosing this problem. there should be a diagnostic flow chart you will need to follow to get to the bottom of this problem. we can help with the terminology and test procedure but you will need this information to get to the bottom of your problem without wasting time and money.
Alldata has a ten day free trial
http://www.alldata.com/landing-page/alldata-free-trial
The radio staying on with the drivers door needs further explanation.
I feel this may be covered in the owners manual. which can be downloaded for free via this link.
http://www.pontiac.com/owners/owner_resources/manuals.html
Before you get too carried away you need to check a few basics, check your owners manual to insure you will not need a code for the radio once you disconnect your battery.
If you do you can buy a inexpensive device that uses a 9 volt battery to keep this radio code active and other computer settings by powering up the 12 volt on all the time power outlet.
http://www.amazon.com/gp/product/B0...emory saver&qid=1451662334&ref_=sr_1_2&sr=8-2
this should be covered in your owners manual or service manual I provided a link to either.
Once you know your safe to do so, disconnect the negative battery cable then the positive cable and check for corrosion clean the battery and cable connections as needed also remove and clean and jump connections found under the hood.
reconnect the positive connections first then the negative connections.
With a DVOM digital volt ohm meter test the charging system you should get 13-14 volts while running.
http://www.amazon.com/INNOVA-3320-A...1?ie=UTF8&qid=1451662609&sr=8-1&keywords=DVOM
You may want to make sure your key fobs have good or new batteries in them, check either manual for reprogramming them if necessary
Once you know you have a good charging system a good battery good connections, no codes in the computer and no warning lights and properly functioning Key fobs with fresh batteries we can begin diagnosing your problems.