You can, but there are gift tax and other ramifications (for example qualifying for medicaid) that need to be considered before you do. Please contact a local elder law or estate planning attorney to be sure you take ALL the appropriate things into consideration.
There are many different ways you can provide such a gift, but you need to do it carefully to protect yourself!
Yes you can. But the best way to do it depends on if he is living in the house and how long he's lived there, or if he will use it as his primary residence. If you need Medicaid within five years, then the gift might disqualify you from getting Medicaid for a while, but there are ways you can minimize this risk. The answer also depends on whether you want to give up total control of the home, or keep half of it, or something in between.