Regarding your first question, while there is a small element of those reading your application realizing that the EC activity was more recent, I wouldn't stress over the difference of a few months. I would suggest applying when you feel you have your application ready, you are not rushing those essays or recommendations, etc, and are happy with your GMAT. And applying early decision can be an extra small plus as it shows your utmost dedication to that program. Either way, you will want to do what you can to demonstrate your passion for and rationale for that new activity you've gotten involved in.
Regarding showing interest in the school other ways, it certainly can't hurt and could possibly help, so if opportunities arise, I suggest you take them when you can-- school visits, info session attendance, contact with students/alum, etc. It also all helps you to truly get to know the school inside and out, and you never know when you'll stumble across a specific nugget of info or insight that will either further solidify to you that the school fits best for you; or that you'll even be able to mention in passing in an interview for example, or both. You can really dig into getting to know the school if you speak with current students/recent alum, which can be very helpful--think about what q's you can ask them, such as 'what would you do differently if you started the program now', 'who is this school right for, and who is this school NOT right for', 'how do you see the school evolving in the coming years', 'how would you describe your relationship with student/professors', etc.
Good luck to you!
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