Who [do we think [ _ had a key]]? ; Do we think [who had a key]?

Question

1. “Who do we know had a key to the place?”,
   
2. “Do we know who had a key to the place?”

When I came across sentence #1 from ‘the Silkworm,’ I thought #2 would be a possible structure. (If they denote the same sense, I’m not sure.) But when matrix verb were ‘think’, can the second-pattern structure (#4 below) be possible?

3. “Who do we think had a key to the place?” (I can see this type in CGEL,p.1082)
   
4. “Do we think who had a key to the place?”

The sentence itself is plausible (you can think a fact, but think a question?) , but the meaning is somewhat strange from #3. Is #4 possible? Possible but not have the same meaning as #3?

Answer 1

4 is not plausible in my opinion as it is grammatically incorrect. The question itself is also poorly structured as one may not ask if you know something without knowing the logical conditions that are required to know or not know such a thing. Look at my example and how when the person in question is made known one can proceed to answer a logic based question on whether we think he has a key or not.

Do we think "Tim" had a key to the place?

When you compare it to your phrasing of the same question

“Do we think who had a key to the place?”

It ends up being phrased in a manner that does not resolve the logical prerequisites needed to be able to answer the question as that is two questions according to the logic used in the question "think who" so you are asking if we think X is Y and who asks us if we know the person behind X is Y. This sort of logical phrasing fails to get the question across therefore it is not plausible.

"who had a key to the place"

as the word, do, ends up questioning if we know or we do not know.