Here are the symptoms:

(I just replaced the battery, thinking it couldn't hurt since I didn't know when it was last replaced. Didn't seem to help)

Sometimes, the door won't unlock when using the keyfob. Have to manually unlock the car.
Sometimes, the alarm will sound immediately after locking the car with the keyfob.
Sometimes, the drivers side door window will not keep the programming and have to be reset.

Recently it has trouble starting, when I turn the key some dash lights come on but it won't do anything. I'll try the key again and it might fire up no problem. Yesterday, it wouldn't start at all and made some clicking sounds non-stop AFTER I took the key out of ignition. I had to turn the key over again a few times and remove the key and it finally stopped clicking. That prompted me to go get a new battery in hopes it was just a weak battery.

This morning the door would not unlock, the window would not go down half inch, didn't start the first key turn, and then started no problem after I turned the key a second time.

Thought it may be loose connections on the battery. But I did tighten them with new battery installation.

 

Take your car to AutoZone, PepBoys or the place you got the battery and have it checked. Did you clean the cable connections prior to installing the new battery? Did you size the battery properly?

 

There are also a number of battery and ground connections forward of the battery that could be causing an intermittent electrical issue. The first being the high amp fuse block in front of the spare tire in the trunk. Had the spare out? JTIS has drawings of the major B+ and ground connections.

Another issue could be the security and locking module which appears to be potentially common to all the issues you identified.

 

I have seen this happen occasionally after a battery disconnect and reconnect. Try disconnecting the battery for a few seconds and reconencting it again. Then test everything. If it does not work try again, repeating the cycle of disconnecting , waiting, connecting and testing over again several times. The system reset in these cars is sometimes a bit flakey.

This of course assumes your battery and conenections are good as mentioned above.

 

The problem was a loose bolt on the drivers side fuse box, creating resistance and the strange behaviors. Tightened it up and car works perfectly It's important that all the fusebox bolts are tight.

 

Kudos to you for finding the problem and posting the what you found!