Learn Russian: Ответа до сих пор нет.

While с тех пор means "since that time" (from then), до сих пор means "until this time" (up to now, or "still"). Notice how the preposition с (from/since) pairs with тех (those/that), and до (until) pairs with сих (these/this).

What exactly are the words сих and пор in this phrase?

This is a frozen phrase from older Russian. Сих is the genitive plural of an archaic word for "this" (сей), which survives in everyday words like сегодня (this day) and сейчас (this hour). Пор is the genitive plural of пора (time or season). Literally, the phrase means "until these times".

Why is the word for answer in the genitive case (ответа)?

It is required by the word нет (there is not), which expresses absence. In Russian, sentences expressing the total absence or lack of something always require the missing thing to take the genitive case, even when it appears at the beginning of the sentence.

Could I change the word order and say До сих пор нет ответа?

Yes, that alternative is perfectly natural. By putting Ответа at the very beginning of the sentence instead, the speaker establishes it as the known topic, emphasizing that the answer specifically is what is still missing.