Heartbeat develops at 6 weeks so you're right at the threshold of whether you'd see a heartbeat because you're not exactly sure of dates. I've had a scan at this point and seen a small (think 5/6mm across) bean shaped thing with an even tinier but speedy electrical impulse inside it (the heartbeat).
You'd expect to see a gestational sac in the right place and then whether you can see the fetal pole, actual baby, heartbeat etc depends on the equipment and how far you are along.
Be prepared for the scan to be transvaginal too. I've had abdominal scans at that stage but they've then done transvaginal ones for a clearer picture.
General anaesthetics are okay in pregnancy btw as I'm sure you've been reassured but I'm no expert.
My friend had a scan at around 6 weeks after some bleeding and a heartbeat was detected via a vaginal ultrasound, although I don't think it's detectable on a normal ultrasound until 7-8 weeks. I think they just check that everything is measuring as expected for the length of gestation.
I had a transvaginal scan at 6 weeks, which I wasn't expecting at all. If you do have one, it's much easier than a smear test (which I find quite painful). I did see a heartbeat on mine but I was quite lucky. It just looked like a tiny blob on the screen with a pulsating bit. Good luck, and take a book for the waiting room as they usually have quite a few people to see.
Just to say I had an early scan and was a bit less along than I thought based on symptoms so only saw a sac and a very small yolk sac and what the sonographer described as the beginnings of a fetal pole (baby) but just looked like a black dot. She said a couple of days either way can make a massive difference to what you can see even trans-vaginally as it's growing at 1mm per day at this stage. I think what I'm saying in a very long, round about way is don't worry too much if they don't find a heartbeat or you can't see much of a baby because a couple of days at this stage can make a dramatic difference. I say this only because I worried myself silly for a couple of weeks till I had a rescan when everything was fine!