Can I plant a variety of herbs in the same big planter?

The herbs will do fine so long as your planter has adequate drainage (support it off the ground with bricks or some other way) and you're using suitable soil. A mixture of equal parts of John Innes no 2, peat-free multipurpose compost and horticultural grit would retain enough water yet have the drainage you need, and you could add a few water-retaining granules and some fish, blood and bone fertiliser as well if you liked. Feed the plants with organic liquid fertiliser, diluted to half-strength then use it at each watering. And of course, keep the herbs trimmed and bushy, doing regular pruning if you find you're not using them for cooking very fast, so they don't get all sprawly and untidy. I expect you've got a nice sunny place for your planter? It sounds an excellent project.