Sometimes I don't get my period at all during my "off" week. The official Nuvaring stuff that comes with my prescription says to test for pregnancy if it happens twice in a row. I have tested for it before, just for peace of mind, but I haven't gotten pregnant and the test always reads "not pregnant", so it's not being affected by the ring/hormones. Also typically it takes me until Thursday to get my period (I take it out on a Sunday), so that 2-3 day thing isn't necessarily the case. And your period will probably be lighter than ever before when you do get it. It's no big deal, I sort of consider myself lucky for not having to bleed for such a long time like before I started using birth control. Also you mentioned feeling pregnant. Have you ever been pregnant before? If not, it could just be the power of suggestion. Sometimes the mind is very powerful, and you might just feel that way because you think you're pregnant. If you have kids, then I guess you know what it feels like and I'm not trying to offend or patronize.
since it is your first ring, you cannot rely on just the ring. My OBGYN gave me the ring and warned me to use other methods of BC (excluding diaphrams) because the ring will not go into effect until a couple of weeks. Hope this helps!