It seems like you were able to start the car after it died on you the first time? If so, I doubt it is a battery issue or alternator.
I would recommend you don't purchase any other parts until you get a professional diagnosis. An Infiniti dealership will charge at most an hour of labor as a flat diagnosis fee; sometimes they are nice and charge half an hour labor.
Depends on whether you let them do the work, or just want the diagnosis report.
A dealership will also possibly address any open recalls on your vehicle. Talking of which, you can check for any recalls by yourself using your VIN online -
https://www.nhtsa.gov/recalls
There were 3 main recalls for these vehicles the one of interest being the gas pedal recall. So even if that turns out to be your issue when you replace it yourself, if you can get it done for free, why not?
Good luck