Yes. You can run multiple campaigns. Many of us do: e.g., exact-same campaigns with only changes in the ad text, to see what does better (and with the idea of eventually terminating the lesser-working ads).
How the ad-placement algorithm works with near-identical ads from the same advertiser is a murky topic.
They don't compete against each other, if you mean a bidding war. So you won't pay any extra if you're running multiple ads for the same keywords, but one ad might take away impressions from the other. I think AMS favors the better performing ad, so if you're trying new tag lines for old keywords, you might have trouble getting enough impressions. Make sure the bid amount is exactly the same and you should get some.