The only definition i can find is that it expresses irony/rhetorical question, but that doesnt really fit into expressions like どういうものか? etc.
So what can be the usages of ものか?
In どういうものか, this もの is just "thing", and no special grammar is involved here. Just as どういう本か means "What kind of book is it?", どういうものか simply translates to "What kind of thing is it?"
There are various special usages of ものか, too. Generally, you should understand ものだ (which by itself has verious meanings) first and think of ものか as its question version.