Is the Heathrow Express free of charge within Heathrow?

yes, no charge when I was there 2 years ago to do that

There's a free shuttle to T-4 from T-5 as well with a 10 minute walk to the Hilton. Or you can take a Hotel shuttle right to the hotel for £4.50.

Bus routes 482 and 490 do the journey directly from stop 7 also free of charge.

Yes, the public buses are free (subsidized by the govt in exchange for stopping the hotels from offering shuttle service). No need to pay to take the Hoppa bus.

Yes, as long as you get off the train before it leaves the airport zone, it is free.

Other options if you want (but probably won't have to since your hotel is connected to the terminal even though it is quite a hike):

The big red double decker busses are free within the airport zone. It can be confusing on which one gets you close to your hotel. I have used them to get back to the terminal when flying out because the Hoppa buses can be very crowded in the morning with everyone trying to get to the terminals.

If you take the Hoppa Buses which run directly to all the hotels, it is £4.50 per person each way and I think well worth it if you have lots of luggage since the regular busses don't go up to the hotel front doors.

Thanks, all.