Oh my god, that chain. clean and lube it, links might be sticking. Don't replace it, can almost guarantee that you'll need to replace the cassette if you do.
It could just need adjustment, I think to start a clean and lube on the chain, cables, shifters and derailleurs is in order.
If the hanger is bent, you aren't fixing it without a tool. It's also pretty easy to see if it is bent, if you have the tool.
The way that cable goes through the derailleur looks wrong, looks like it could bind, probably wouldn't cause the shifting issues though.
Edit: also verify chainring and cassette/wheel aren't bent (lift bike, rotate wheel, cassette cogs shouldn't wobble too much), spin crank backwards, make sure no chainring wobble.
It could just need adjustment, I think to start a clean and lube on the chain, cables, shifters and derailleurs is in order.
If the hanger is bent, you aren't fixing it without a tool. It's also pretty easy to see if it is bent, if you have the tool.
The way that cable goes through the derailleur looks wrong, looks like it could bind, probably wouldn't cause the shifting issues though.
Edit: also verify chainring and cassette/wheel aren't bent (lift bike, rotate wheel, cassette cogs shouldn't wobble too much), spin crank backwards, make sure no chainring wobble.