Paul Broussous has already addressed the question in the title, and since it seems that you are in fact more interested in the last two questions, I will focus on these.

I don't think there is a treatment following steps 1) and 2) available in the literature, but it should be possible to work this out since the representations of $\mathrm{GL}_2(\mathcal{O})$ are known explicitly. On the other hand, it is not clear how useful this would be, because already for $\mathrm{GL}_2$ over a finite field $\mathbb{F}_q$, this is not the preferred approach to the representations (apart from the principal series, of course). This is partly because it quickly becomes unmanageable, and does not allow for an inductive approach in the same way as parabolic induction.
Moreover, over a finite field even step 1) is problematic in general, because the irreps of $B(\mathbb{F}_q)$ are related (via Clifford theory) to the irreps of the upper uni-triangular subgroup $U(\mathbb{F}_q)$, and the classification of the latter is known to be a wild problem (for large enough p, one can describe the irreps of $U(\mathbb{F}_q)$ using a Kirillov orbit method formalism, but it is not clear that this would be helpful for the problem at hand).

Also, in step 2) there is going to be a lot of repetition, that is, two induced representations may have irred constituents in common, and there may be no easy way to tell when this happens and for which irred constituents.

Finally, as you mention in one of the comments, the approach via steps 1) and 2) can only work if there is a good description of the $B$-$B$ double cosets, and this is not available in general. Note however that this is not the only place where wildness enters, as the above remarks on the situation for groups over a finite field shows. One advantage of the approach via Clifford theory and orbits (following Hill), is that one can avoid wild classification problems by restricting the construction to a subcategory of representations, such as the regular representations.