Can a private or lowest ranked person in the army (or navy, air force, etc) be demoted and kicked out for an infraction?

If you are at the lowest rank, you cannot be demoted, but, you can be kicked out.  I have seen it happen lots of times.  And, you can be kicked out at a higher rank as well.  I saw a Sergeant get kicked out for alcoholism - 6 months before he qualified for a pension.

they can kick anyone out of the military for most anything

doesn't have to be a pvt

Oh Yea ..................................dishonorable discharge !!

depending on what you did, yes. or they might just put you in jail and you will have to make up that time when you get out.

Private is the lowest rank. You can't demote a private.