The stems are supposed to turn purple. They can flop over too, if they get too long. If you cut the stems at the lower nodes, it will branch out and strengthen the base. Keeping the soil moist might also help to keep it upright.

The darkening of the stems is related to the plant getting more light. It's normal. Probably, at the place that was selling the plant, it was kept for too long on the shelf without enough light and developed these long bendy stems.

Definitely cut the plant back and water it well. Mint grows back VERY fast so don't be afraid to cut it way, way back.

What sort of mint is it? Could it be one which has running stems anyway?

When I bought it the stems were tall and straight and green. They flopped over when I started keeping it on my window sill.

Mint tends to like partial sun (4-5 hours per day) - check how much light they're getting on your windowsill and check how much water you're giving it - Mint likes moist but not wet roots. Most mints will spread - and this can be a great thing if they're in pots (not so great in the ground though) as if you put a little soil over part of the stem that is hanging down, it'll root and spread. They also like a good liquid fertilizer every now and then (it works wonders at bringing your mint back after a caterpillar attack).