You can get symptoms early on, way before an HPT would tell you positive because of the hormones that the luteal cyst gives off in your body before the placenta takes over. A luteal cyst forms when the egg bursts out of the ovaries. It is benign and it shrivels and dies and disappears way before baby is born!
You can google "luteal cyst" and see what comes up! Good luck to you!
I guess it would be possible. I do know that it takes at least 5-6 days after intercourse for the egg to actually implant into the uterus. Typically, I think most of the pregnancy symptoms you experience are because of the HCG rising, which might not happen that soon. Also, sometimes its hard to determine between pregnancy symptoms, and PMS symptoms. For me personally, there were many times I thought I was pregnant because of symptoms, but it turned out I wasn't. You might be able to get a pos HPT at about 9-10 DPO. Anyways, if you are TTC, good luck!!