Sort of. You can have a current sanction card from one league and bowl in another, non-sanctioned league. Your scores in the non-sanctioned league will not be recognized by the USBC, and you will not be eligible for any awards.
However, Bowling in the non-sanctioned league will not invalidate your USBC sanction, and you will be eligible for any awards earned in a sanctioned league.
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Yes. While a union may prohibit union workers from working non-union jobs, in the sport of bowling, there is nothing that prohibits a sanctioned bowler from participating in non-sanctioned events.
However, if a bowler in a non-sanctioned league wishes to bowl in a sanctioned tournament, there may be rules that prohibit the bowler from participating, or require the bowler to purchase a sanctioned card and/or require, if a tournament that uses handicap, to use a higher average or not receive handicap.
Most basic fouls would be bowling out of turn, crossing the foul line and using the wrong lane :-)