Okay, several things.
1. Pregnancies are dated by your LMP. That's just how it's done, until you can get a better estimate based on ultrasound. Your doctor isn't wrong for doing it that way.
2. Based on your ovulation date and cycle length, yes, you are probably closer to 5 weeks than 6 weeks. It is likely that you'll go back in a week and see a heartbeat. That's good, so take a deep breath! When you go back, if there is indeed a fetal pole and hb, they will modify your due date based on the measurements they get from the ultrasound, which is the most accurate way of doing it. But since they don't have accurate measurements yet, they will still be going by your LMP.
3. Your doctor wasn't wrong to prepare you that you might be miscarrying, either. One sign of a miscarriage can be a lack of fetal pole and heartbeat when you'd expect to see one (and he did expect to see one based on your LMP -- which again, is how pregnancies are dated). Like I said, based on your cycle length and believed conception date, you are probably okay, but miscarriage is still a possibility and it's better to be prepared than not.
4. Pick up a few books about pregnancy. It'll really help you understand this stuff so you don't have to freak out quite as much. I like Your Pregnancy Week by Week but PPs have given good suggestions as well.