How to loosen traditional faucet stem?
As I look at the pics, it appears s if there is pitting and corrosion.  Maybe it is time for a new faucet.  To use the current one, I would saturate it with a corrosion cutter, repeat about every 30 minutes.  There is a clip in the top picture that needs to be removed.  A screwdriver will generally remove it.  This will allow the stem to be removed.  I would soak the nut again, and use an adjustable wrench, an open end wrench or a box wrench.  Then go primate on it.  I suspect once it gives you will be on your butt wishing you only went half primate.  Good luck, hope this helps.
I bought a plastic faucet at Lowe's 20 years ago for less than $10 for an apartment that I owned.  It still works, never have had a repair done to it!