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There are definitely many questions about the usage (or lack thereof) of か in
casual speech. Although it is relatively common in anime, it sounds harsh,
maybe rude in everyday casual speech.

Alright, understood. My question then is: when _do_ we use it? I know of
embedded questions, such as:

> 何を買ったか知っている？
>
> "do you know what they bought"

But I'm not interested in those. I know of 何か and 誰か as distinct words, but
I'm not interested in those either. I think I've heard things like:

> あの女の子が僕を好きなのかと思ったけど

Where the か is in a quoted clause, and I am interested in those, as well as
things like:

> できるか

Which I understand can mean something like "as if I could do it".

Basically, besides embedded questions, when might we actually use か in casual
speech?

Edit: although sundowner's answer is helpful, I'd also like to hear maybe a
more informal description, with examples, of when casual questions use か,
since sundowner's answer only says that questions can take か.
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usage particles particle-か casual
Using か in casual speech
OK, foreigners' grammar does not seem to focus much on each particle per se.
This is more or less a translation of the dictionary definition linked in the
comment.

Sentence-ending か could mean:

  1. Question marker
  2. Question marker, with implied negative.

  * どうしてそんなことしなければならないのか Why on earth do I have to do it (I don't think I have to).

  3. When denying the interlocutor's assumption

  * そんなこと知るか How am I supposed to know that (= I don't know!)

  4. Let's

  * ご飯食べようか Let's eat

  5. Urging (mostly appears as ないか)

  * さっさと宿題やらないか Why not start working on your homework (The speaker is urging to do the homework)

  6. Surprise, exclamation.

  * 誰かと思ったら君だったか I was wondering who it was, and ah, it was you.

The distinction between 2 and 3 is a bit fuzzy.

か in かとおもう is still a question marker, but practically it means _maybe_. E.g.
your sentence means _I thought maybe the girl liked me, but_.

できるか could be a simple question (rising tone) or means 3 (decreasing tone,
usually stressed). Particularly in a manga, it is a common scene where A tells
B to do something impossible and B says できるか!, which means _How could I?_

* * *

Hopefully the following clarifies your question.

Consider the following

  1. きのう何を食べましたか？
  2. きのう何（を）食べました？
  3. きのう何（を）食べた？
  4. きのう何を食べたか？

As you say in the comment, 4 is weird. But note 2 is possible in ます form. Also
in 2 and 3, を tends to be omitted, which is possible in 1 as well.

On the other hand, all of the following are possible.

  5. きのう学校に行きましたか？
  6. きのう学校に行きました？
  7. きのう学校に行った？
  8. きのう学校に行ったか？

5-7 are common with decreasing politeness. 8 is possible, but sounds strongly
masculine, like from someone superior in some sense. I don't know a proper
English word, but in this case for example a father can say 8 to his child.

One possible explanation is that non-ます-form + か sounds masculine, and in case
of wh-question like 4, it sounds almost military. For example, an officer
could say to subordinates この非常事態を何と心得ているか _What do you think of this
emergency?_
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