Cleanup: Refactor CodeClimbersCore

[Tanner-Scadden]

I think we can clean up https://github.com/CodeClimbersIO/browser-extension/blob/main/src/utils/CodeClimbersCore.ts a lot now that we have things wired up and working.

We should reuse the request util taken from CLI, and I think we can simplify a lot of this logic.

[YashrajKupekar17]

I would like to solve this issue. Please assign it to me.

[Tanner-Scadden]

If you have any questions just let me know here or on discord!