Q1. 今天没有下雨 expresses the weather condition that it did not rain today.
今天不下雨了emphasizes the consequence that the rain has stopped.
Q2. 俩 means two, is almost equal to 两个,just much more casual than 两个. so yes, it is used much more often in Spoken Chinese.
e. g. 我们俩去看电影吧?
这俩人真有趣;
这俩词意思一样;
你们俩不要再淘气了 ...etc.