I believe so, unless there is an alternate reason [other than age] why you can't.
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Yes, you can.
NO you can only join the military at the ages of 17(with a parents permission)-36 years of age.
Yes.
Yes, but it all depends on the severity of the crime you commit, felons aren't allowed in, and there is possibility to get a waiver to let you join the military as well.
If you are free of the cancer for more then 5+ years, you are allowed to enlist.