How can I propagate my lucky bamboo?
You can take cuttings, dip the cut into rooting hormone, and them put the stem into soil. After a few weeks, the stem will start to root. Many people do this without root hormone for successfully rooted cuttings as well.

Does the stalk still grow after its been cut?
The stalk will not get taller. It will, however, sprout new shoots most likely near the top of the plant. Those shoots will continue to grow taller.

How long do they live?
Much of the answer to that question I suppose is up to the individual grower. I have read up to a few weeks or even many months. Mine has lasted for a couple of years and is still going strong. I am also expecting many more years from my lucky bamboo plants.

Another answer I have read is that lucky bamboo plants will last indefinitely. That almost sounds like longer than ourselves at least. Perhaps they last forever if they are great master lucky bamboo.

;-)

Oh yes... you can root the cuttings in water too.

Here is a link that might be useful: How To Reproduce Lucky Bamboo

My lucky bamboo is growing in the wide vase. Water from this vase vaporizes very promptly. I am going to be on 6 weeks vacation. How can I save my bamboo for long time?

Have a friend take care of it while you are away.