If I still am in the early action group (july 1-october 15), does submitting July 1 give me better odds of submitting October 15?
http://admissions.tamu.edu/freshman/admitted
Based on the description of the process, if you are qualified for automatic admission but are applying to a competitive major, it is best to apply early, since most majors are filled on a first come first served basis (but some have additional criteria). If you do not qualify for automatic admission, you may end up waiting for a while until they know how many non-automatic-admission applicants they are able to admit (but the more competitive majors may have been filled by then).
Another consideration to keep in mind is the earlier your are accepted into Texas A&M, the earlier you can submit your housing deposit, which will make hard to get into options such as Hullabaloo a possibility for you.
Not really (at least technically) but you shouldn’t be submitting October 15th. You should be safe at submitting around September, and there is no rush to do July 1st (no advantage compared to the few months after). Don’t wait until October 15th, because something could come up and you can easily get screwed.