Does the phrase "It is he who came to my house." sound natural?

Only if you are talking like yoda or a Jedi 

Although this is grammatically correct, this is a very old-fashioned way to speak and would not sound natural. This type of conversation would be from years ago.

This is he who came to my house.

It sounds natural to me. However, to most it wouldn’t, but it is grammatically correct. It’s like when people say “I think it was him” instead of the correct “I think it was he”.

However, “He is the one who came to my house” is also grammatically correct.

It's grammatically correct, but schools have stopped teaching that.  Instead, you would hear:  he is the one who came to my house.

Not in American English. It sounds upper class. We would say:

    " He is the one who came to my house".

Though your way is correct also.

You could also say, "He is the one who came to my house."

For almost any purpose I would say, "He came to my house".

No, it doesn't, yet ironically it is the "correct" form.  It was he who came to your house.

Sounds a bit theatrical or poetic.