According to the rules of golf, are you allowed to play a match play and a Stableford match at the same time?

For example, if you are a member at a golf course, can you play on men's night (which uses a modified Stableford scoring system) and play a match play at the same time?

What happens in cases where a penalty is assessed and, in stroke play it's a 2-stroke penalty and in match play, it's a loss of hole? An example would be grounding your club in a hazard... I would think you would be penalized two strokes in your Stableford match and your opponent would win the hole in the match play, correct?