Rent from property in other State

Yes you have to obtain another registration in hinmachal

No not required .
You can raise bill from Delhi by charging IGST .

As per section 12(3) of IGST Act, the place of supply of service in relation to an immovable property shall be the location at which the immovable property is located.
Therefore, in this case, the place of supply is undisputedly Himachal because the property is located there at.

Note : Yes in case any expenses incurred in relation to property in Himachal , you cannot claim ITC for the same , because the POS will the where the property located