Migrate to ESLint 🔧

Created by: felixfbecker

Closes #2461 🎉

Very excited about this one as ESLint has support for a bunch of new rules that we can now check for that often came up in code reviews, e.g. referencing this.state inside setState() and many more.

I recreated our TSLint config in https://github.com/sourcegraph/eslint-config and made https://github.com/sourcegraph/tslint-config/tree/eslint contain only the few TSLint rules ESLint doesn't have.

Migratingwas more work than expected but it's a huge improvement to our codebase, e.g. finally refactored all usages of async-unsafe componentWillReceiveProps() to componentDidUpdate(), removing unused state properties, ...

Also like the better configurability of ESLint - configuring rules for glob patterns or overriding severity in comments.

I also noticed we were not typechecking our code in CI! So this fixes it.

cc @JoshuaKGoldberg