Using か in casual speech

Question

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.

Answer 1

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?

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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?