Regarding your last, non-highlighted question: the operation you describe is called Hopf stabilisation, and it correspond to doing a connected sum of $\theta$ with a contact 3-sphere (the binding is connected-summed to a Hopf link -- whence the name). Therefore, on the manifold level this doesn't change anything. Let's see what happens to the contact structure.

If the Dehn twist along $C$ is negative (or left-handed), the Hopf link is negative (_i.e._ the two components have linking number -1), and the contact structure on the 3-sphere is overtwisted (with $d_3 = +1/2$, if I recall correctly) and the resulting contact structure is overtwisted, with the same Chern class as $\theta$ but with a different $d_3$ (shifted by $\pm1/2$ -- I need to check this).

If the Dehn twist along $C$ is positive (right-handed), the Hopf link is positive (_i.e._ the two components have linking number +1), then the contact structure on the 3-sphere is the standard one (the only tight one), and this doesn't affect $\theta$ at all. I think that this is also called a Giroux stabilisation.

Giroux proved that two contact structures given by two open books $OB_1 = (S_1,f_1)$, $OB_2 = (S_2,f_2)$ are isotopic if and only if $OB_1$ and $OB_2$ are stably equivalent, that is if and only if they have a common (positive) stabilisation.

I could go on an on about this story for quite some time, but I think that Etnyre's [_Lectures on open book decompositions and contact structures_][1] contain a lot of information.

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Regarding Stein fillability, a contact structures is Stein fillable if and only if it has a supporting open book whose monodromy can be factorised as the product of positive Dehn twists. If you want to know more about fillability and Stein surfaces, Ozbagci and Stipsicz's _Surgery on contact 3-manifolds and Stein surfaces_ makes for a really pleasant read (in addition to organise a lot of results, proofs and references). For planar surfaces and genus-1 surfaces with connected boundary, there are even stronger, very recent statements which are not contained in the book.

[1]: http://people.math.gatech.edu/~etnyre/preprints/oblec.html