Is it possible to notice spotting two days after unprotected sex? And when people say "it's light" I don't understand whether they mean its very little or the color is light. And can it be enough to fill up half a pad?

Yes, this is possible. Especially if sex was rough or you are near your period.

When they say "light" they mean the amount. Yes, filling up a half a pad can be very common.

So would a HPT be accurate if I was to take it after the spotting stops? Or should I wait another week or so?

No it would not be accurate until you miss your period. It takes a couple of weeks for the pregnancy hormone hCG to build up in your system enough to be detected on a pregnancy test.

Well, I have always had strange periods so having a period monthly is not common for me. I had my period in September & then October so I wasn't expecting one this month. It's also hard to keep track of my periods because they seem to just start on random days. It's never the same day for every month. It gets even more complicated because I've had very light periods before when I was younger. They were light throughout the whole 5-7 days & look similar to my situation right now. So I seriously don't know what's wrong with my body.