Add more formatter rules and enable TS linter in Biome

## Description

Thanks to @rbm0524, we replaced prettier with biome (#5820), a Rust-based, super-fast JS/TS formatter and linter.

Currently, we are only applying very basic formatting rules in Biome, but I think we can have more rules to improve the formatting of our TS codebase.

Also, as a biome can serve as a linter too, introducing linter rules would help us check common human errors automatically.