This is fine, but take note that 正在 carries the meaning of "she is (doing this) right now".
Is 洗澡 like 跑步, first character a verb and second character a noun?
Yes, you can think of it this way. Normally verbs like this are intransitive.
Should 她现在洗澡 be accepted or is the second 在 required?
It is required.
It is not required. If they wanted the English to reflect the necessity of the second 在 they should have said, "she's in the shower right now"
What is the grammatical reason for this?