Are these parts compatible?
 

Hello, I'm planning on upgrading my Schwinn Katana and I've decided to upgrade this bike due to the majority of the parts failing on me or performing abysmally. If anyone could tell me if these parts are compatible, that would be nice and pointing me in the right direction would also be a plus, Thanks!

There's a lot of hills where I am also.....
Crankset: SHIMANO SORA FC-3403 (Planning on getting)
Pedals: PD-R540 (Planning on getting)
Cassette: Shimano CS-6500 (Planning on getting)
FD: Shimano Ultegra FD-6700 10 Speed Front Derailleur (Purchased)
RD: A050 rear derailleur (not upgrading this yet)
Bicycle Chain: KMC X8.99 Bicycle Chain 8-Speed (Compatible with 6/7/8) (Purchased)
Brake Levers: Shimano BL-R400 (Already have)
Shifters: Shimano A050 Handlebar Mount 7 Speed (Already have)

IDK (by the part numbers) .. are the shifters and the cassette the same 'Speed' Count?,
then your odds improve..

I'm sort of mixing and matching and pretty much all the parts are different speeds..... I haven't upgraded the cassette yet but it's going to be a 9-speed. The shifter is 7-speed only.

7 is not 9 is it?. think !

only way they will work is to dump indexing entirely and use a friction shift lever .

want 9 speed stuff ? you need a 9 speed cassette compatible freehub,

a 9 speed Indexed shifter , a 9 speed chain , 9 speed cassette ..