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I had a successful interview today and start in a couple of weeks. However, I also plan on sending my Army Reserves application off in a couple of days. Due to the number of days this requires you to be off, one big 2 week block , I'm not sure if I should mention that I've applied when I start.

Should I hold off my application for a couple of months first and settle in first?

I would hold off anything that might be disruptive to the crucial time when I'm new in a job and making the impression with my colleagues and bosses that will be the most lasting one.

You want to go in looking, keen, committed, and happy to be there, not passing time between personal stuff. Give them a chance to at least remember your name.

Tell them once you've been accepted and know the dates you'll be gone.

If you haven't even applied yet, then there's no guarantee you'll be accepted. (Correct me if I'm wrong and you know they will accept you.) At this point, there isn't anything to tell that's helpful to them, especially if you aren't sure of the training dates yet.

If you are confident that you will be accepted and you know the dates, then yes, you should tell them when you start. If there's regular training in addition to the initial two weeks, then you should provide a calendar of that to them, too. I would make sure you know your rights in your country first, though, in case they try to pull any illegal shenanigans.

It depends.

If you are joining the National Guard or the U.S. Army Reserves, you will be protected by federal law from prejudice and mistreatment by any employer. Still, you may not want to give the perception of having misled your employer. That said, you may not want to tell them that you might be gone, perhaps even out of the country, at the discretion of people in Washington D.C. who have pencils behind their ears. And you may not want to mention national emergencies.

As a writer above has recommended, wait until you actually have a date for Basic Training/AIT/weekend drill. Keep in mind that some units will want you to attend drill before you attend your initial training.
Most drills take place on the weekends. Will you be working on the weekends? Is there anyone else at your prospective job who is a weekend warrior?

On the other hand, you may want to inform your prospective employer that you are going to join up. If the training directly applies to your prospective job, that could be a huge feather in your cap. Some military jobs have training which is extremely expensive on the civilian side. Another important factor is whether you will get a security clearance. This can also be of vital interest to a future employer and make you much more valuable.