There is probably no definitive answer to a question like this, but here's more information with a different take on some topics.
그를 얼마나 좋아했는지(를) 나는 모른다. -> I don't know how much I liked him.
그를 얼마나 좋아했던가(를) 나는 모른다. -> I don't know how much I liked him.
They are both correct but the first one sounds much better.
-지 (는지, -ㄴ/은지, -ㄹ지) is a verb form intended for a mid-sentence noun clause like this. It is very flexible in this function and can create a clause with little positional restriction.
-는가 is primarily intended as a sentence ending (the dictionary lists it as 종결 어미 (clause/sentence ending), whereas -지 is listed simply as ending which includes both connective and sentence-final ending).
-는가 can be used like -지, but it is generally a second choice. It is a little more formal than -지 and has a feel of asking a general, open question to people at large rather than a specific, personal one. -지 on the other hand is limited to in-clause use and can't serve as an independent sentence ending. Sometimes people use -는가 in place of -지 to create the more formal and impersonal effect, because -지 sounds familiar and ordinary.
As for your individual questions.
- Are there any differences in terms of their semantic implications?
=> As was mentioned, the difference is more in nuance and formality.
- Are there any differences in the type of clauses (in terms of tense, aspect, e.g.) they select?
=> In present/past tense, they conjugate the same; in future they diverge.
- 사고의 원인이 무엇인지/무엇인가 밝혀야 한다 (present)
- 왜 그런 결정을 했는지/했는가 모르겠다 (past)
- 동생이 올지 확실치 않다 / 동생이 올 것인가(는) 확실치 않다 (future - no simple construct for -가)
(There is another similar ending -ㄹ까 which can be used here, as 일찍 올까(는))
- Are there any differences in terms of what case markers or delimiters can follow them?
=> Generally not. But in future tense -가 becomes more roundabout as shown above.
There are a few idioms that can work well with -가 though.
- -는가 보다 = It appears that ... e.g. 강아지가 아픈가 보다 = It appears the puppy is sick.
- -는가 싶다 = I have a feeling that ... e.g. 속도가 너무 빠른가 싶다 = I feel we're going too fast.
(This 싶다 expression works with a lot of different verb endings including -지, 나, -다 and others)
- Are there any differences in terms of what case markers or delimiters can follow them?
Do they both require a wh-word in the embedded clause? For example, below are circumstances in which there are no wh-word in the embedded clause, and I am not sure about their grammaticality:
=> As for the wh-word, both -지 and -가 work similarly. If there is no wh-word, they take on the meaning of whether it is true of not.
- 그가 나쁜지/나쁜가 나는 모른다 have the same meaning, but again, 나쁜지 is much more natural.
In simple everyday sentences like this, -지 is by far the best choice.
