Yes, so if you have a half double and rent out one side while living in the other side, you would deduct 50% of the mortgage interest and property taxes and the other 50% on Schedule E.
Depending on the type of renovations you do, you would either expense them or add them to the cost basis of the property. You would allocate the cost based on the part of the property renovated. Example, if it was a half double and you renovated the rental side but not your side, the money spent on the rental side would all be allocated to that side. If you did something like granite counters on your side, then that would be strictly personal.
When you sell, yes, you could only claim 50% as your home, the other half would be sale of your rental property.
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