If you keep the watermelon moist and away from direct sunlight it can last for about a month in good shape, however if the sun dryes out the watermelon you'll get 2 weeks tops, as for mold I've never had a issue with it and if you do its probably because you're adding too much water and it would've happened in any pot.

If you wish to plant using the watermelon you can get some smaller melons and make holes on them to make it easier for moisture to reach into the soil