Can i hold a rail ticket from basingstoke to Westdrayton but get out at the exit of London Paddington?

The expectation is that you would change trains at Reading to one of the stopping trains that run between Reading and Paddington and call at West Drayton. Your ticket most certainly is NOT valid to travel beyond West Drayton, and if you were to get on a non-stop train you'd be treated as having no valid ticket beyond Reading. If you want to travel on to Paddington on your return journey then you both need to previously buy a ticket from West Drayton to Paddington, AND travel on a train which actually stops at West Drayton (the rule in the National Conditions of Carriage about combining tickets is that unless one of the tickets is a season ticket then the train must actually call at the place where one ticket ends and the other starts).

If you travel back to West Drayton and don't get off there you will have to pay for a ticket to cover the ride from West Drayton to Paddington. The guard on the train will charge you the full single fare for that journey but if you buy a West Drayton-Paddington ticket in advance it may cost you less. Check the fares on-line (www.nationalrail.co.uk) and you may find that an advance purchase day-return between West Drayton and Paddington is a lot cheaper than a single bought on the day of travel. If you have one of these you'll be able to return to West Drayton too (if you go before midnight). Don't chance it by travelling on to Paddington without a ticket. They nearly always check tickets these days. You'll get stung for a big fine, and still have to pay the fare!

No, you can't stay on the train to Paddington. You may receive a Penalty Fare for doing so. Buy the ticket from Paddington to Basingstoke instead - then it will be valid for West Drayton - Basingstoke and Basingstoke - Paddington.