Sometimes when a preposition meets an article, they have a little wedding, and here are the results:
- in + la = nella
- in + le = nelle
- in + il = nel
- in + i = nei
- in + lo = nello
- in + gli = negli
So if you wanted to use the article, you could say Lui entra nella mia casa. But if you wanted to just stick with in, entra in mia casa is a bit strange.
There are a few locations that can take a or in without an article, in ways that are not always possible in English. Siamo in biblioteca (we are in the library) andiamo a chiesa (we go to church) lavoro in città (I work in the city or in town) devo andare in banca (I have to go to the bank)...but you can usually use the prep+article combination for these anyway, people have just started to drop the articles over time.
While this is not incorrect "enters my house", no one would ever say it this way. WE would say he goes in the house, or he comes in my house. The whole sentence should be scrapped really. At most we might say he enters the house. This would be used if someone, like a policeman, was describing a sequence of events. But its a very formal and unnatural way of speaking english for most people.