CI: Add Biome Linting, Formatting, and Husky Pre-commit Hooks

[Ashrockzzz2003]

This issue proposes migrating our code formatting and linting infrastructure from Prettier to Biome. Biome offers a unified toolchain for formatting, linting, and more, potentially simplifying our development setup and improving performance. This migration will involve updating our CI pipeline and pre-commit hooks (via husky).

https://biomejs.dev
https://typicode.github.io/husky/

[priyansh-narang2308]

/assign

[Devpool-soc]

Alright @priyansh-narang2308! Consider yourself officially assigned as now. If you are thinking, "Whoa, hold up, I didn't sign up for this," no worries, just yell "/unassign" and we can pretend this never happened. But if you are sticking around, just know your clock for this ticks out on: 2026-01-15. Don't screw it up :)