1. Yes, You can surely add your mother as a Co-owner while purchasing the property.
2. Yes, You can claim the benefit of Housing Loan solely on your name, and any excess benefit left out can be claimed in the name of your mother.(if required)
3. Preferably not, as it will be too difficult to prove and convince the tax authorities and would involve unnecessary hassles.
4. If You are not showing rent paid to mother, then you cannot claim tax deduction of HRA but can only claim interest paid to bank as housing loan.
Trust this clarifies your query.
Feel free for any further clarification.
CA Rohit R Sharma
BCOM, CA, LLB-GEN, CERT FAFP.