How is the Heathrow Express with a large bag going into Paddington? Is it necessary to purchase the tickets in advance?
Thank you

There is luggage storage at the end of the compartments, you can buy a ticket on the train or at a ticket booth or machine.

DON'T buy the ticket on the train (they charge extra).

There's a bank of machines just before the train. Or you can even buy them from a (generally queue-free) manned ticket kiosk.

The storage space mentioned makes it fairly easy to handle a large bag; you may want to sit nearby to keep an eye on the bag. You don't have to buy the ticket far in advance; just get it (as described in the other posts) before you board the train. Another option is to buy the ticket online (www.heathrowexpress.com has details) and pick up the tickets from a machine at the station or airport.

Those racks are multi-tiered and have plenty of room for larger bags. Very similar to the racks you see on some rental car courtesy shuttles.

I'm not sure security is really that much of an issue unless someone snatched your bag and jumped off the train either before it left the airport or once you arrived in Paddington (since there are no stops in between).

You may as well buy the ticket in the machines at the airport; I paid for them on-line this past February and still had to collect them from the machine which is very easy to do.