Car will randomly shake and sputter on both idle and acceleration and dome light flickers like alternator is bad even after new alternator installed. What could be potential cause of all these problems coil pack/plug maybe a bad ac pump causing excessive resistance?

 

Likely not an AC issue but AC is and can be a heavy load on an engine.

Is the Done Light OEM filament or an aftermarket LED Type. ..?

For the Misfire..Check the #5 Coil Connector for any broken wires.. Then Pull the Coil out and check it visually..

Inspect the Boot visually for cracks.. Then pull it off the Coil and Check the Coils Pintle..( the Point inside the Coil) often times I find Corrosion here or just burnt up period from a cheap coil.

I would then pull the plug and Check its porcelain for cracks. It will appear as a Black Line you cant wipe off.. Hello Carbon. Start there to verify and rule out a spark issue. Dont be surprised if you find the tube full of oil, if the VC's have never been done.