I will respond under the impression that you're primarily interested in contact 3-manifolds. Much more is known in this low-dimensional case as techniques such as Dehn surgery etc. can be adapted (with some slightly non-trivial work) to work for contact manifolds in this dimension.

1) One easy way to construct contact manifolds is using the Thurston-Winkelnkemper construction. For dimension-3, this construction asserts that if you have an open book decomposition of your manifold of interest $M=M^{3}$, then that decomposition uniquely determines a contact structure on $M$. This decomposition consists of a link $B\subset M$ and a fibration $\pi:M\setminus B \rightarrow S^{1}$ such that the boundary of every (compactified) fiber $\pi^{-1}(\theta)$ is equal to $B$. I suggest you check out John Etnyre's lectures on open books for some background on this: There you can see that every oriented, compact 3-manifold has an open book, and so a contact structure. This follows from the Lickorish-Wallace theorem, asserting that every compact, oriented 3-manifold has a "nice" surgery presentation. For example, every fibered link in the 3-sphere determines a contact structure on it via this construction. These appear "in nature" when you look at Milnor fibrations.

In higher dimensions, things don't work out so nicely. You can still use open book decompositions to build contact structures, but you need additional hypotheses. Each fiber must be a Liouville domain, and the diffeomorphism of the fiber associated to the open book (which makes sense to discuss as $M\setminus B$ is a mapping torus) must be a symplectomorphism. I guess you can Google the relevant definitions if you're interested :)

If you're interested in less constructive methods of determining whether or not a manifold admits a contact structure, then there are currently not many tools available. There are the characteristic class obstructions to the existence of an almost contact structure on a given $M$ described by Geiges, but little is known as to when an almost contact structure can be deformed to a contact structure. Etnyre (see his most recent paper on the ArXiv) figured out how to do this in dimension 5 -- building upon work of Presas et al -- but it seems nothing is known in higher dimensions.

2) It seems like there are extreme restrictions on what non-vanishing vector fields on a 3-manifold can be the Reeb vector field associated to a contact form on it. A few years ago, Cliff Taubes proved the Weinstein conjecture which says that every Reeb vector field on a closed, oriented 3-manifold has a closed (AKA periodic) orbit. On the other hand, there are lots of examples of vector fields on 3-manifolds without closed orbits. Examples are easy to write down on the 3-torus and there are (highly non-trivial) examples of such vector fields on the 3-sphere (which are counter-examples to the Seifert conjecture). It is expected that this conjecture holds for Reeb vector fields on contact manifolds of all odd dimensions (every Reeb vector field has a closed orbit).