"I know that he is watching me."
Translation:Je sais qu'il me regarde.
IIRC = If I recall correctly.
Imperative tense in French = http://french.about.com/od/grammar/a/imperative.htm
Regarde-moi = Look at me
Here is an excellent explanation of the different uses of connaître and savoir. http://french.about.com/od/grammar/a/savoirconnaitre.htm
I looked this up a while ago. I think the technical reason is "connaître" is only ever used with a noun phrase
Je connais Marie Je connais New York Je connais son poème Je connais ce livre etc
Anything else, you use savoir. And "he is watching me" is a clause, not a NP. If anyone can explain why this is the case it'd be interesting, because it seems like a bizarre rule to me.
No you can't. "Moi" is a stressed pronoun. The rules for stressed pronouns are rather complex. To learn about them, see here: http://french.about.com/od/grammar/a/pronouns_stressed.htm