Getting a tube train after an overnight flight from the US with three bags between two of you, will be more than okay. I did it alone after an overnight from Australia, with two bags.
I wouldn't bother with the Heathrow Express/Connect unless you are staying near the station ie. as in walking distance.
One problem with arriving earlier to your hotel, unless you have booked an extra night you won't get your room until the afternoon. Drop off luggage and start your sightseeing, although you feel tired and grotty.
Advantage/disadvantage of airport shuttle buses - they can take a long time, as you drop off other passengers along the way. It is a way of filling in time along with getting to see a bit more of the area you are staying in. Personally I find it boring and takes forever.
Years back, we booked a car from Addison Lee to take us from North London to the airport - from memory it cost around £50, guessing it would have gone up quite a lot since then. Thinking about it, the last time I stayed in Paddington got a taxi from there to Heathrow and it cost us closer to £60.

