Best Way to Upgrade My Bike
2200 is the Shimano Claris line from 2009, so not 'unbranded', but at the bottom end of Shimano hierarchy.

Firstly, the derailleur itself is probably not causing shifting issues. Replacing the shift housing and cable will very likely fix the shifting and keep the indexing properly adjusted. Check chain and cassette wear as well.

A 9 speed Sora derailleur with a 8 speed cassette and Claris 8 speed shifter should in fact work. The reason is that all Shimano road rear derailleurs and shifters at 10 speed and below generally use the same cable pull ratios. You can also consider the Claris RD-R2000, which will work a little better than the RD-2200. 

The RD-R3000 and RD-R2000 will work with a cassette with a large sprocket up to 34 teeth.

The 48/38 crankset is a bit weird. Fitting a smaller ring depends on the bolt circle diameter (BCD) of the crank. What has probably happened is that a previous owner has fitted a smaller big ring to a 130mm BCD crank which limits the small ring size to (usually) 39 teeth. If the BCD is 110mm, you'll be able to go to a 34 tooth small ring at least.

Remember that if you do change the cassette you'll need to replace the chain as well.

Edited to add: There is another route to a drivetrain with a spread of ratios - buy a different used bike that has what you want (and re-sell the current bike). This is my standard advice in these situations as upgrades can easily cost a significant fraction of the original cost of the bike, and entail headaches and frustration.