Does every element have a half-life? Will every element slowly decay into something else? Or do some isotopes have infinite half-lives?
 A: The stability of a nucleus depends not just on the number of protons that it contains (which determines its element) but also on the number of neutrons. Atoms of the same element that have different number of neutrons in their nuclei are called isotopes. Some isotopes of an element may be stable and will never naturally decay (as far as we can determine); other isotopes may be unstable.
According to this Wikipedia article there are 80 elements with at least one stable isotope, and a total of 252 stable nuclides. The element with the largest number of stable isotopes (10) is tin.
A: In agreement with the commenter, this question is a bit ill posed and should probably refer to the nuclides, rather than elements. There are in most cases a large number of different isotopes each of the same element. Additionally, rather than asking if it has a half-life, it would be more appropriate to ask if it is stable. I'll take a stab at it anyhow, given that I think the meaning of the question is more or less clear.
Stability is somewhat in the eye of the beholder. Do we consider it stable if the half-life is longer than the observed universe? For example, there are some rare decay modes such as double-beta decay which have very long half-lives (perhaps $10^{25}$ years, compared with our universe's measly $10^{10}$ years). These nuclides are certainly considered stable by everyday standards, however, upon closer inspection, do technically have a slight possibility of decaying (and therefore have a half-life).
Take a look at the NNDC's table of nuclides. Here, the default coloring is for half-life, and all the stable nuclides are shown in black with >$10^{15}$s half-life. Some of them simply read "STABLE", and to the best of our knowledge, don't decay, however, again, this may also indicate a very very long but known half-life decay mode (pick your favorite double-beta decaying isotope like 76Ge or 130Te or 136Xe).
For another look at the same data in a different view, check out the isobar view on the IAEA data website. Each isobar forms a mass parabola, as you would get from the semi-empirical mass formula (SEMF). If you have "too many" neutrons or protons you can decay inwards towards the bottom. In general, there is always only a single stable nuclide per isobar, and even on the even-even isobars with two points of local stability, double-beta decay may be allowed at a very low rate. Even if a nuclide is stable against beta decays or electron capture (EC) and sits at the bottom of an isobar, it can still decay by other methods, such as alpha decay, or cluster decay, or fission.
For the lunchroom colloquial discussion, most practicing physicists consider these exotic and long-lived decay modes to be "stable", although it is important to recognize that these are options at times.
A: 
Will every element slowly decay into something else? Or do some elements have an infinite half-life?

Hydrogen is an element, the nucleus of which is made up of a single proton.
The proton has never been observed to decay and a current estimate of the half-life of a proton is at least $1.67\times10^{34}$ years.
So, for all intents and purposes, hydrogen's half-life can be considered infinite.
