Both of these are grammatically correct and the choice of which one to use would depend on context.
I see, I see could mean anything from an absent-minded murmur of agreement to an irritated "Yes I understand, stop going on about it."
I see. I see. The slight pause between them could mean you are acknowledging true understanding of something which is being explained to you. Or it could be angry, as in:
"I see. I see. You think you can behave exactly as you like and I'll still here be waiting for you! Well I won't!"
My house, my rules is a statement that I am in charge because I pay the bills.
My house. My rules is more emphatic, perhaps repeating it to someone who has broken a rule.
My house! My rules! is a full blown argument often followed by "And if you don't like it you can pack your bags and get out!"