The pre-trained ones no (except some of the new ones which have post training data added to pre-training for some reason). The post-trained ones yes (at least all the ones I've seen).

Some of the niche ones I'm not sure about. Like the historical LLMs. I have not tested those yet.

I think any instruction tuned model is going to "know" it's an LLM.

Yes. The first step of aligning each and every GPT-based LLM is to suppress the “I am human” kind of responses. It’s baked into the weights.

Without a system prompt no. And in general they “know” nothing and just predict the next best word.

https://en.wikipedia.org/wiki/Original_face