If I’m planning to take the AMC12/AIME (in Feb.) or the state math contest (in Mar.), can I still submit more scores to the colleges that I applied to?
If not, is it still a good idea to spend extra time preparing for it (P.S.: I do enjoy doing these math problems)?
Yes, for sure. It is fine to send additional EC accomplishments or awards to them. Email the general admissions email with the info, ask them to please consider it with your application, and reiterate your interest in the school.
I doubt they will see a March result. Admissions to MIT come out every March 15 (Pie day). UChicago and WUSTL are early too. Ivy Day is by April 1.
The only exception is if you note it on your application and are a math award winner of some repute (state/national) then they may care but in that case they would expect you to take them. Besides the first two schools are EA, so apply early and this will be moot.