Enable strict mode in Typescript

As discussed in #2119, enabling [strict](https://www.typescriptlang.org/tsconfig#strict) mode in the Typescript compiler would reduce the chance of certain types of bugs occurring.

Running `tsc --strict` reports 321 errors. Trying to fix these errors in one PR would be a bit daunting. Instead, there might be a way to gradually enable strict checking on a per file/directory basis. There doesn't seem to be a Typescript-official way to do this, but [this package](https://github.com/allegro/typescript-strict-plugin) looks like it could be helpful.

Start by running `tsc --strict` and reviewing the 321 reported errors. Read issue #2119 and evaluate the linked typescript-strict-plugin as possible guidance for incremental checking by file or directory. Done means strict checking is enabled through an agreed approach and the reported errors are addressed.

Written by the indexing model from the issue text.