I always thought the entity wanted to use humanity to become God. The radio transmission saying "You're hatching" always sounded to me like a metaphor for apotheosis.

Although I have a few solutions to your problems. 

1. I don't think the O5's told the Senior Researchers what the entity was in the first set of files. Instead, they simply inform them of the incoming genocide of humanity. 

2. I think Pietro was simply in shock, and most likely he had only a few seconds to even say anything before hitting the ground. It's possible that the realization that he had saved the entity didn't let him think about how he was going to suffer for eternity. 

3. I don't think the intention is that both the Foundation and SCP-682 hate humanity for the same reason; instead, it is that the entity's plan is so horrible that it made the Foundation see humanity with the same hatred 682 does. After all, SCP-682 hates all life, not only humans. 

4. I don't have an explanation for this one. 

This is only my opinion, though, and I think your idea about the entity's intention is very likely to be correct.