Is there enough hydrogen left after a star dies so another star will have enough to light up? A star consumes quite a lot of hydrogen in its life, and is pretty much "vacuuming" everything in its vicinity. After it dies (eventually by supernova which will spread all its composition over light years), is there enough hydrogen left in that area to light up a new star? And will that star be more short-lived compare to its predecessor?
There are several misconceptions in your question.

First, a star does not vacuum everything in its vicinity. Rather it forms from a condensation in a gas cloud, which in turn collapses to a proto-star surrounded by a gas disc, which can contribute further material. Once formed in this way, a star typically does not acquire more gas (exceptions are symbiotic binary stars etc).

Second, a star with mass in excess of $\sim8$M$_\odot$ will (typically after a long time) suffer from a supernova, when most of its envelope is flung back into space. That gas is still mostly hydrogen, though enriched by 'metals' (non-primordial elements). However, the gas is hot and fast moving and hence not in a condition to form another star. 

Third, the gas from the supernova will eventually mix with other gas and dissolve into the general pool of interstellar medium (ISM). Some of that may cool to form a molecular cloud (as gas cloud where molecular $H_2$ dominates), which in turn may become the site of new star formation.

We know that the Sun has been formed from enriched material, which is a mixture of primordial gas with the ejections of several supernovae.