baby will still be moving slightly up and down. But if you think about it, its hardly going to be an accurate measurement is it, they have to try and work out by feel alone how far in your babys head is and i doubt thats easy.
as for your BP what did they say? usually its the second figure that they would be concerned about and 80 is not high at all. rushing about from one appt to the other may have cused the rise so i expect they will just keep an eye on if it there were no other signs of problems.
I really feel for you in this heat. I was heavily pregnant in the last really hot summer we had in 2005 and it was unbearable, my legs would itch and id sit at work with all the fans on me eating one ice lolly after another.