We don't know for sure.

There's two paths this could take.

1. They did not, because they weren't expecting to die at such a young age.
2. They realized they were in the middle of all out warfare and writing a Will was recommended.

Let's choose path one first. They didn't write a will, and assumed that if they passed that baby Harry would live with Sirius. (Not Lupin, who they suspected to be the spy.) They also probably assumed the house would be left to Sirius to live in until Harry came of age to take ownership. Remember, they probably thought if they were to die, it would not be directly at their home, but probably on a mission somewhere. They also probably thought one of them would die if it came to it, not both. If one passed, obviously all ownership would belong to the surviving.

But anyway, if they didn't write a will, all ownership belongs to Harry and would be kept by Sirius until Harry came of age.

Path 2: Probably something similar. Again, they weren't expected to BOTH pass away, they didn't expect to die at their home, and they didn't expect for Sirius to be taken to jail. So, they probably would have left their house and money to baby Harry, to live in with Sirius until Harry turns seventeen and can take ownership for himself.

So, probably the same goes for both!

We don't know for sure.

There's two paths this could take.

1. They did not, because they weren't expecting to die at such a young age.
2. They realized they were in the middle of all out warfare and writing a Will was recommended.

Let's choose path one first. They didn't write a will, and assumed that if they passed that baby Harry would live with Sirius. (Not Lupin, who they suspected to be the spy.) They also probably assumed the house would be left to Sirius to live in until Harry came of age to take ownership. Remember, they probably thought if they were to die, it would not be directly at their home, but probably on a mission somewhere. They also probably thought one of them would die if it came to it, not both. If one passed, obviously all ownership would belong to the surviving.

But anyway, if they didn't write a will, all ownership belongs to Harry and would be kept by Sirius until Harry came of age.

Path 2: Probably something similar. Again, they weren't expected to BOTH pass away, they didn't expect to die at their home, and they didn't expect for Sirius to be taken to jail. So, they probably would have left their house and money to baby Harry, to live in with Sirius until Harry turns seventeen and can take ownership for himself.

So, probably the same goes for both!

Final answer is there's nothing in canon, so we don't actually know.