Well, according to this site, the SRAM GX type 2.1 has a chain wrap capacity of 47T. So it should be fine if you replace with a longer chain. If it has the same cable pull ratio as your old 10 speed shifter, then you won't need to change the shifter. SRAM calls the ratio branding on this derailleur "Exact actuation 1:1".
https://www.worldwidecyclery.com/pro...railleur-black
A little bit cheaper in Germany 50€.
https://www.bike-discount.de/en/buy/...g-black-449072
If you're interested to learn how to service a bicycle, you could do all of this yourself for the experience. It's not difficult, maybe 1 hour.
$50 for the new derailleur, $15 for a KMC 10 speed chain, $5-15 for a new hanger (you have to find out which one works with your frame).
3) Changing the derailleur just needs a hex wrench. You would have to cut the old cable end to get it on the new derailleur with a cable cutter, and crimp on a new end cap (pliers are good enough for crimping). If you don't change the shifter, you don't need to change the cable. Takes maybe 5-10 minutes to install.
4) 5 minutes to figure out how long of a chain you need, 5 minutes to break the new chain with a chain breaker and install the new chain.
1) A lot of hangers are produced by Union. They sell them OEM to the bike brands. You can simply buy the hanger directly from an ebay or amazon seller. I got mine for $5. Change the hanger. 5 minutes.
MARWI | Gear hangers - Dahon
2) You would have to bend the hanger into place. I bought the hanger alignment tool for around $25 on sale. It takes some time to learn the first time, but it's pretty simple. 10 minutes for the first time. And it's handy to have the tool in case you hit the derailleur out of alignment in the future, and to check it once in a while.
https://www.bike-discount.de/en/buy/...ent-tool-25599
5) then when everything is installed, check the cable tension and low and high stops. 5 minutes.