This is a little out of my wheel house. what would be the appropriate way to try to remove the I don't know what you call a sleeve or whatever that is stuck on the I guess you call it the caliper mounting post

It is called a brake boss and that looks like a sleeve from the inside of a V-brake or Linear brake. Squirt it with some penetration oil and then use a small pair of channel lock pliers to twist it off. It also looks like your brake bosses are replaceable, I think I see flats to get them our, most likely a 9mm open end wrench would work, may have to heat it up first as there is normally thread locker of the bosses threads.

By the way, if you tap on the phone screen on the object you’re trying to photo it usually forces it to adjust the autofocus, it had decided you wanted a pic of the wheel there. You might have been slightly too close to that sleeve thing though, for it to be able to.

Yeah the bike is a Schwinn Phocus 1500. Promax V-brake TX119.

I didn't realize I posted that other pictures of the picture I meant to post.

Still focussed on the wheel rim / spoke nipple

Macro shots are tricky

Grab it with pliers and give it a wiggle. If it resists thread a long M6 screw fully into the boss, so you don't squish it, and escalate the pliers force, but be mindful that the steel is only 1 mm thick. Worst case, you can file through the brass sleeve to loosen it. Then apply a thin film of grease when fitting the brake arm, so it doesn't happen again (the brakes will also work better).

If simple penetrating oil and pliers don't do the job the OP might have to file away the brass sleeve along one lengthwise point and then peal the sleeve away. I've seen the steel post rust so badly that the brake arm (and that sleeve being tightly contained in the arm) become too tight to slide off. If the steel boss unthreads from the base one can use a M6 bolt and nut for a handle. Thread the nut on the bolt and the bolt in the open end of the boss and now this handle can be held by hand or in a bench vise. Andy

​​​​​​ actually my focus is only on getting my bike fixed and that first photo it was not detailed enough to see what needs to be seen and second photo does the job so but I appreciate your critique though😎

well I borrowed your file idea or the concept anyway. I didn't have a file but I had like a cone shaped grinding stone in my drill.

The thing about using a flat file is that you can take off a neat slice from the sleeve until you get down to a thin line of contact with the boss underneath, doing minimal (hopefully zero) damage to the steel.