Trying to change the air filter on 2001 Chevy Cavalier

Is the engine receiving fuel? Check fuel pump output and fuel line pressure.
Check HT lead and coil connections.
Check spark plug condition.
Check air filter condition.

New spark plugs, oil and oil filter, transmission flush and fill, fuel filter, air filter, coolant flush and fill, clean throttle body plate and bore, clean the MAF sensor, replace any loose or cracked vacuum lines, PCV valve, new air filter, new battery.

99% of the time all you should ever need for a tune up is new spark plugs, and a clean air filter and throttle body. But since your inquiry indicated "major" I gave it more critical thought.

If it has a big black plastic bolt head and cover at front middle of car on engine then remove that cover. To make removal easier take off air filter inlet tube and move the wires close to cover. The filter snaps into cover so you should be able to pull it right off.

it sounds like a dirty throttle body. Just spray clean with throttle body cleaner. Or it could be a plugged air filter...not getting enough air. Or possible fuel filter needs to be changed.

remove the gas tank, undo the ring that holds the sending unit , replace fule pump and new sock filter, reinstall

no it should not be vibrating, Have you tuned it up. Changed the fuel and air filter. this should clear it up

Many filters come with an internal check valve inside...chances of getting two that are defective are pretty slim. You are positive you have the right filter on there? Chevy used to have a check valve right in the filter mount pad, have not paid attention to that in a while but if you see a small round orifice with a little metal cover in it, you may want to push gently on it with the back of a ball point pen to free it up.
If the pad is held on by two small screws, you may want to remove the pad and check for gunk buildup inside. If not, try filling the filter with Marvel oil and putting it back on. It may dissolve whatever is blocking the bypass. You don't have to remove any oil, 1/2 qt over will not hurt anything. If it dosen't work you will need to use compressed air to blow through the passages. You have been driving without a filter (it's in bypass mode) for probably a few months already another few days won't hurt anything especially since you have new oil in there.

In most cases it is the fuel filter. I don't know when it was changed before, but it should be changed at least every 8-10 k miles. Make sure you change the air filter too.

get a friend with a working spedo on the phone driving next to you talk about your speeds

There are a bunch of problems there,from clogged engine air filter,fuel filter that needs replacement to a transaxle problems;start with the most easy-to-solve items and so on,consulting a GM car care manual can be a good help too.luck!