I have a question on why my car vibrates when it’s parked. I thought I was trippin’ … I’ve been learning about my infiniti cause I’m fixing it myself. I don’t trust mechanics so if I can get any advice what to check, I’d appreciate it.
Check my spark plugs, the mounts, && what else too check …?
Also, I have a problem with the fuel gauge. I took out the two fuel pumps in the back to clean the sensors, but how do I know if the pump(s) are damaged && need to be replaced?

 

There's steps in the Factory Service Manual section DI - Driver Information to test the floats, the primary pump harness you check resistance across pins 2 and 5, the sub pump is just two wires so read resistance across them. Resistance should scale 3-40 ohms on one and 4-80 ohms on the other iirc, you can work the float arm by hand and see if resistance changes appropriately.

As for the vibrating, sounds like it's a misfire, listen at the tailpipe and see if it's arrhythmic, if so it's slightly misfiring. If it sounds like it's holding perfect rhythm then it's probably something else. You can HEAR the pulses from each cylinder firing.

You can sort of check your motor mounts by just opening the hood, start the car, put it in drive, then put it in reverse, switch between the two. Watch the engine through the gap in the hood, does it shift A LOT? If so you might have a broken or worn motor mount, it will always shift a little bit on stock rubber mounts so a tiny bit of movement is normal.

If you're on old spark plugs, or if you don't know how old they are it's probably time for new ones. They're only good for about 75k miles or so before they start having issues. Pay attention to any oil in the spark plug well tube, scorching or damage to the coil pack well tube boot including the contact spring inside the boot, blistering of the epoxy resin on the coil pack top (black shiny part).

Also see if all the plugs are mostly the same color or if you have a couple that are darker/lighter/have more deposits. I like to use a sharpie to write on everything as I remove it, write on the top of each coil pack, write on the ceramic insulator of each spark plug with what cylinder number it is.

Pay close attention to your coil pack wiring harnesses, it's easy to get a couple of them swapped, use a wrap of tape and that sharpie to label everything.

Torque everything to factory spec, spark plugs and coil pack torque is under section MA - Maintenance in the FSM. (coil packs are actually EM-35, there's a link under the appropriate section in MA).

https://www.nicoclub.com/infiniti-service-manuals