Hi, sorry for your loss.
In answer to your questions
- If you have not lost all products of conception, then you may well be recommended to have an ERPC operation. Did you have a scan following the medical management? Have you taken any pregnancy tests since this time? I have had two miscarriages where the hcg has taken 5 weeks to reach negative point. So your current positive pregnancy test could well be left over hcg from your previous pregnancy. You need to get blood tests and / or scan to determine if this is a new pregnancy or not.
- Yes you can fall pregnant quickly after a miscarriage. But with your dates it would have to have been very quick ovulation and implantation, and unfortunately not very likely. I did have a faint positive test 3 weeks after a previous miscarriage (and produced my last child), but it is not normally that quick for most people.
With it being so soon after a medically managed miscarriage, I would strongly recommend that you get checked out quickly. As if you do have retained products, then you are at risk of infection. It may be that your hcg is taking a long time to go down, this does not mean you can not ovulate in the mean time. I hope you get an answer soon.