Car doesn't seem to get enough gas to engine

Check the idle position sensor.

Have you done a tune lately on the car?
I would also suspect a fuel pump problem is you've done a tune up recently.

I HAD SIMILAR PROBLEM MY 2005 PT SUDDENLY JUST SEEMED TO CHOKE DOWN AND PUSHING ON GAS PEDAL MADE IT WORSE SO I LET UP ON PEDAL AND THEN SLOWLYMPH ACCELERATER BUT I WOULD ONLY GO ABOUT 50 MPH AND THE ENGINE LIGHT CAME ON I CHECKED OIL IT WAS FINE I THEN PUT A CAN OF SEA FOAM IN GAS TANK AND DROVE SLOWLY A WAYS THEN SPEEDED UP IT RAN FINE SEEMED TO BE DIRTY INJECTORS OR FUEL PUMP. I WNT ON HOME THE ENGINE LIGHT STAYED ON THE NEXT MORNING THE ENGINE LIGHT WAS OFF AND IT HAS RUN FINE EVER SINCE

A variance of 20 PSI indicates a problem.

Might be a modual needs replacement

your battery may not be fully charged yet with the quick jump. let it run for a bit with no lights or accesories, no a\c or heat. everything is run off the computer and it need to be fully charged.

could be a few things the start up could be the idle air control sensor (my first thought), the coolant temp sensor for the computer might be a secondary thought for the engine being cold if the computer thinks its warm. that would be the first 2 places i would start without a check engine light or code to reference

2001 elantra gets 300 miles per tank of gas regular unleaded
slow to respond when accelerated 105 K original miles recently purchased
no check engine lights most parts look original engine revs up but tranny seems slow to respond?

Your noise sounds like it is power train related,If you have eliminated the engine,Since it doesn't make the noise while moving and in neutral.
It sounds like a universal joint in the drive shaft.Do you notice a vibration
along with the noise?

Higher octane gas 93 with some good octane boost in the black can at autozone cost about 7 dollars or a bottle of 104 octane boost. Change the fuel filter. Probably only one coil pack is bad. The OBDII reader should tell you which one was a misfire.