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 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.