Bike re-assembly gear and noise issues I had an old general purpose bike that I wanted to use exclusively with my new turbo trainer. I ended up taking it apart almost completely and of course I did not take pictures so re-assembly was a pain and still have issues now.

Gears don't really change in many positions and result in different types of noise. One gear position worked fine so I  just used that. Now however after about 200km cycling over a couple of months it started to make a noise almost like metal sparking. It's not constant but very frequent and gets worse with higher speed.

Cleaning, oiling, using WD-40 did not resolve the issue. Neither did inflating the tire (it was very low psi). I don't see any obvious wear from friction or other damage. The axis at the center of the wheel gets really hot after 10 min at least not sure if this was always the case and if it's normal.

Can anyone see any issue in the pictures? What else is worth trying to fix this?
First thing to do is check the rear wheel bearings. Get the rear wheel out and inspect the axle. If there is any play between the axle and hub or roughness or 'notchiness' when you turn the axle, the bearings are worn out. If you have a cup-and-cone style hub the bearings can be serviced. There are several online resources that show you how to do that. I recommend Park Tool's articles and YouTube videos. Some hubs have replaceable cartridge bearings.

If the gear shifting is not working, that could be due to a number of different things, including the issues with the shifter, but most likely poor indexing adjustment. Again, there there are number of online resources that show you how to troubleshoot and adjust indexing.

I note you have an older Shimano Tourney, which is is lowest level Shimano groupset, which does not offer great shifting in the first place.