Not all hypersurfaces in $\mathbb{R}^{2n}$ are of contact type.
Weinstein, in the paper: "On the hypothesis of Rabinowitz' periodic orbit theorems", where he defined the concept of contact type, gives a criterion. If $H^1(\Sigma)=0$, and $\Sigma$ is of contact type, then the characteristic line bundle comes with a distinguished orientation, determined by those vectors $\xi$ such that $\alpha(\xi)>0$ for all contact forms $\alpha$. This is independent of the contact form. For periodic orbits this induces a positivity criterion. In the same paper he also constructs an hypersurface in $\mathbb{R}^4$ which is not of contact type, by showing that the criterion is violated.
Many hypersurfaces are of contact type, as you remarked. Another nice example are mechanical hypersurfaces. These are hypersurfaces arising as level sets from hamiltonians $H=T+V$, where $T$ is the kinetic energy term $T=\frac{1}{2}\vert p\vert^2$, and $V$ is a potential depending only on $q$. This works in general cotangent bundles.
A partial answer is as follows.
In order to make use of the theory of holomorphic curves (with boundaries or asymptotics, whatever), we should restrict ourselves to symplectic manifolds with convex boundary, or more restrictively, Liouville domains or Weinstein domains. These domains then admit completions into Liouville/Weinstein manifolds by attaching cylindrical ends. (Strictly speaking, the notion of a Liouville manifold should mean something more general, and even symplectic manifolds which are of infinite topological type.)
In the case of Weinstein domains, the relation between SFT and wrapped Fukaya categories is well-known. Explicitly, let $(M,\theta,\phi)$ be a Weinstein domain, with $\theta$ a contact form on $\partial M$ and $\phi:M\rightarrow\mathbb{R}$ the Morse function which determines the structure of $M$. Then $M$ can be realized as a handle body decomposition
$M=M_0\cup_{i=1}^k H_i,$
where $M_0=N^{2n-2}\times D^2$ (with its corners rounded off) is a subcritical Weinstein domain (with $N$ a $2n-2$-dimensional manifold with boundary), and $H_i$ are critical handles (in this case, $n$-handles).
For every $H_i$, there is a Legendrian attaching sphere $\Lambda_i$, so all together we get a Legendrian link $\Lambda=\cup_{i=1}^k\Lambda_i$. For $\Lambda$, there is a well defined $\mathbb{Z}$-graded DG-algebra over a semisimple ring $\mathbf{k}=\oplus_{i=1}^k\mathbb{K}$, namely the Chekanov-Eliashberg DG-algebra $\mathit{LHA}^\ast(\Lambda)$. One important result that can be extracted from the paper Effect of Legendrian surgery due to Bourgeois-Ekholm-Eliashberg is that $\mathit{LHA}^\ast(\Lambda)$ is quasi-isomorphic, as an $A_\infty$ algebra over $\mathbf{k}$, to the wrapped Fukaya category $\mathscr{W}(M)$ of the Weinstein domain $M$.
This quasi-isomorphism can be interpreted in another way, in the sense of Abouzaid's geometric generation criterion. Namely the Lagrangian cocores $L_1,\cdot\cdot\cdot,L_k\subset M$ associated to the handles $H_1,\cdot\cdot\cdot,H_k$ together generate the wrapped Fukaya category $\mathscr{W}(M)$.
Now this fact has a number of applications. For example, in the case when $M$ carries an exact symplectic Lefschetz fibration $\pi:M\rightarrow D^2$ (in fact, by a theorem of Giroux-Pardon, this is always the case for Weinstein manifolds), it's natural to take $M_0$ to be $F\times D^2$, where $F$ is a smooth fiber of $\pi$, then the above result can be applied to verify that $\mathscr{W}(M)$ is the localization of $\mathscr{F}(\pi)$ along a natural transformation $T:\mu\rightarrow\mathrm{id}$. Here $\mathscr{F}(\pi)$ is the directed $A_\infty$ category associated to the Lefschetz fibration $\pi$, $\mu$ is the autoequivalence on $\mathscr{F}(\pi)$ induced by the global monodromy of $\pi$.
In the context of homological mirror symmetry, $\pi:M\rightarrow D^2$ is usually mirror to a complete algebraic variety $X$ over a specified field $\mathbb{K}$ together with a (usually singular) divisor $D\subset X$. Denote by $Y$ the non-compact algebraic variety $X\setminus D$, which should be the putative mirror of $M$. The localization result above then reduces the verification of the triangulated equivalence
$D^\pi\mathscr{W}(M)\cong D^b\mathit{Coh}(Y)$
to two (usually much easier to verify) triangulated equivalences
$D^b\mathscr{F}(\pi)\cong D^b\mathit{Coh}(X)$
and
$D^\pi\mathscr{F}(F)\cong\mathit{Perf}(D)$,
where $\mathit{Perf}(D)\subset D^b\mathit{Coh}(D)$ is the triangulated category of perfect complexes. As $D$ is usually singular, it's usually not equivalent to $D^b\mathit{Coh}(D)$.
Conclusion. What I want to say is that most versions of mirror symmetry makes sense only in the homological setting (with the Gross-Siebert program as an exception), namely you should deal with invariants which come from Lagrangian Floer or SFT rather than working on the geometric level directly. Once you group these invariants together, those come from Floer theory and SFT are actually compatible with each other, and in many cases, they are just different ways of understanding the same thing, and combining these two viewpoints together do give you something interesting. On the other hand, if you think directly on the geometric level (say asking what is the mirror of certain surgeries in the symplectic or contact category) is kind of wild or unprofessional, and I don't think this is the correct way of doing things. What you should really look at is the effect of these surgeries on invariants, and try to figure out what kind of operations on the mirror have similar effect, then you would probably get some kind of surgery or operation which would conjecturally fit into the mirror correspondence. There are considerations of this flavor due to Katzarkov, Kerr, etc., they considered what happens to a Landau-Ginzburg model as its mirror goes through a birational cobordism.
Best Answer
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).