Derailleur does not match up with chain after tube replacement I have been recently replacing rear inner tube on my bike, since then I was not able to fix the
wheel back in. 

It makes unusual clicking sounds when backpedaling and one side of the chain gets loose. Upon further inspection I think the problem is that the chain does not properly fit into the rear cogwheels, as if the chain got longer while replacing the wheel?

Also, if I shift to the smallest cogwheel on rear derailleur, the part with two small wheels gets collapsed as if there was not enough tension making in practically impossible to function.



As I said, the bike was working perfectly. I only took out the rear wheel, replaced inner tube and put the wheel back in.

Here is a video of me trying to backpedal. Also notice how the part of the derailleur is wobbling - it is because the chain seems not to fit in the teeth and causes resistance.
I think this was a pre-existing condition in your drivetrain that you just didn't notice before removing the wheel. Have you tried riding the bike and seeing if it functions as before, when the right way up and pedalling forward?

There does seem to be resistance in the jockey wheels in the derailleur cage. You can try removing, cleaning and re-lubricating them.