[#1936] Migrate app.vue to TypeScript

[vvidday]

Part of #1936

Proposed commit message

Currently, despite the addition of TypeScript support, the frontend is
still largely written in JavaScript. This results in a lack of type
safety and many complex objects being passed around as unknown types,
which may in turn lead to errors.

Let's migrate the root component, app.vue, to TypeScript. As this is the
entry point of the app, converting this file to TypeScript will enable
the conversion of other Vue components.

Other information

The diff might look confusing because of the re-ordering of the options (apologies for missing this in #1867), please look at 4bfc3aa to see the actual diff of the TypeScript conversion.

I had to use any to define the app property of window.app - not too sure if it's possible to type this.

Lastly, regarding the topic of combining/separating PRs, I was not too sure whether to separate this from #1937. Because of the nature of TypeScript migrations, the PRs would probably be mostly sequential (e.g. this PR relies on the previous PR for the this.$store to be typed), and hence also contains the changes of that PR. This might result in more work, such as requiring maintainers to merge a specific PR before another, as well as any changes to the preceding PR might have to be merged back into this one. I guess there's a tradeoff between the above and keeping the PR's to 'one small change'. Currently, I leaned towards separating them as each change is fairly substantial and they are two separate files. Would love to get any feedback/opinions on this.

[ckcherry23]

Great job on getting the TypeScript migration underway! I noticed that there are a few objects that could benefit from having explicit types as they are destructured later on. Left some comments to that effect.

[ckcherry23]

LGTM!

[zhoukerrr]

LGTM

[github-actions]

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