Update dependencies to latest versions

[merlinnot]

Description

Issue #439, partial resolution.

This PR:

• updates all dependencies to latest versions
• moves some @types dependencies to devDependencies
• refactors a few functions that did not compile because of improved types
• renames typings field to types, as it's the base name
• changes node_modules/.bin/tsc to tsc in all scripts in package.json
• orders fields in package.json in an alphabetical order, so it's easier to find relevant fields

[thechenky]

I'm not a fan of alphabetized package.json as it was quite difficult for me to find things like name which I'm used to seeing at the top. Additionally, I think it's odd to have devDependencies and dependencies not next to each other. Can we leave that file as is?

[merlinnot]

I changed the order of top-level properties back to original.

[merlinnot]

Closing in favour of #450