"I am in my house."
Translation:Ich bin in meinem Haus.
And in English one would usually say "I'm at home" or "I'm home". Perhaps this was meant to be a more literal statement about being inside the building.
I don't think it's correct. "Ich bin zu Hause" means "I'm home". Your home could be an apartment or even your parents' house .... And you could own a house which is not your home. Therefore your suggestion would not be a good translation for the phrase "I am in my house".
Jaye16, as I understand it, meinem is the dative/neuter form of the possessive pronoun "mein". It ends in "em" because it takes the dative neuter ending. It takes the dative/neuter ending because it follows the preposition "in". See: http://en.wikipedia.org/wiki/German_adjectives#Weak_and_strong_inflection
for some help with adjective endings, which, in this case, seem to match the possessive pronoun endings.