To my knowledge there is no advance discount offered. But make sure you purchase a ticket from the ticket machine at the Heathrow station, as there is a surcharge if you pay for it on the Train itself.

You can take the luggage cart(s) from your arrival terminal all way the into the Heathrow Express track level [ an elevator ride down, down, down)...but are prevented from taking them onto the actual platform. Once you arrive at Paddington there are luggage carts available ( for an exact change fee)....that will enable you to cart your luggage down the platform at Paddington, across the station to the taxi queue. Depending on the time of day the queue can be long....not due to an absence of cabs, but more related to the insane traffic flows in and around Paddington.

Have fun in London.