It's usually worth having more than one bike, if they're sufficiently different - where "sufficiently" is a matter of personal opinion, some people hafe a different coloured one for every day of the week. Myself, I make do with two - one for the summer and one for the winter, a colleague uses ine for commuting and one for training.
Back to the rest of your story - while multiple bikes are definitely a good idea, it's also no big deal to put a set of drop bars on your hybrid. With "tyres" I suppose you mean both wheels and tires - the former are stressed when going over a curb, the latter can be punctured.
Basically, the CX as well as the hybrids I've seen usually have similar width rims (actually, high end CX rims are narrower than hybrid rims), so you should be able to run similar tires on either bike (frame dimensions permitting). The strength of the wheel, unless you run absurdly low spoke counts, is more depending on the quality of the wheel build and the quality of the material (hubs, rims and spokes). As for the tires, you have a choice: lightweight or puncture resistant. The most puncture resistant tires I know about are Schwalbe Marathon Plus - I ride them or their cousins on my bikes whenever posible.