Look in the owners manual, or look up online, the procedure for programming the remote lock key fob. If the batteries went dead it may have lost the digital 'key' that the car needs to recognize it as a proper remote and not a 'stranger'.
If that does not work the reciever may have gone bad or lost power. The receiver fuse may be blown. check ALL the fuses. Sometimes they put it on a circuit that is different from the solenoids themselves. I am not at shop right now so I can not pull up the wiring schematic up to tell you which one it is.
About the power door lock working only in manual mode on the one door; the lock soleniod is probably bad. We see it all the time in any electric lock vehicle. You should pull the door panel and check for voltage at the pigtail. When you press the lock/unlock button and get 12 volts at the connector for the solenoid then it is getting a proper signal and the solenoid itself has gone bad (just replace it). If there is no voltage, then there is a fault in the wiring or possibly the switch.
Shop manuals will NOT give you step by step procedures for electrical issues. It will provide the info you need to troubleshoot a problem. This is one that you just have to take some time to figure out. Luckily it is a rather basic circuit.
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