I don't know enough to answer for certain. The main guy in this article had a green card though, so he wasn't here illegally, but it was revoked when he was charged or convicted. I imagine most of these guys had similar immigration statuses. I don't think an illegal immigrant can join the military. I'm not an expert though so I might be wrong.

No. Illegal entry can be charged as a misdemeanor (8 usc 1325) but rarely is. Illegal re-entry after a formal removal can be charged as a felony (8 usc 1326) but again that's fairly rare and you more see it used in extreme cases where that wasn't the only crime, or in cases of people who repeatedly get caught doing it.

Administrative grounds of inadmissibility resulting in voluntary returns or final/expedited orders of removal are much more common.

If they were unlawfully present (regardless of entry method, so visa overstays and illegal entry) for more than 6 months or more than a year, and then departed the US, that also triggers administrative grounds of inadmissibility that bar later legal re-entry for certain periods of time, but would not be a felony.

No. Illegal entry can be charged as a misdemeanor (8 usc 1325) but rarely is. Illegal re-entry after a formal removal can be charged as a felony (8 usc 1326) but again that's fairly rare and you more see it used in extreme cases where that wasn't the only crime, or in cases of people who repeatedly get caught doing it.

Mostly administrative grounds of inadmissibility resulting in voluntary returns or final/expedited orders of removal are much more common.