I always tie them with ribbon , or whatever you like ,near the top under the petals, looks like a hand tied posy and stops them bending over. You'll need a thinner vase.
I bought my mother in law a bunch of tulips recently and she put a drop of bleach in the water before she put them in the vase. I couldn't believe it as I thought it would kill them. It works I've done it twice now, apparently it kills the bacteria.