Quote by sneutron
1. Is current thinking that strings have to wind "around" something? If so, must the compactified space have at least one hole?

Sure  you need topologically nontrivial, noncontractible cycles, otherwise th e cycle shrinks to nothing.
Quote by sneutron
2. With just a single hole, presumably there can be just one topologically distinct string configuration (ignoring winding number and sense)?

yep
Quote by sneutron
3. With two holes, three string configurations  or maybe more with "twists"?

Usualy one determines a basis of cycles (generators of the homology group), and
considers all other nontrivial cycles as linear combinations of the basic wrappings.
For your question of two holes, the answer is essentially yes, but to be precise, you must specify more data (like if the space is compact or not). For example, a compact Rieman surface with two holes has four basis cycles  two "around" the holes and one "around" each neck. Lets denote this basis by (b1,b2,a1,a2). Then a general string wraps around a cycle of the form n1 b1+ n2 b2+m1 a1 + m2 a2. One says that the string has wrapping numbers (n1,n2,m1,m2) with respect to that basis; these numbers can also be viewed as certain charges.
Quote by sneutron
5. How many holes are favoured, and are there enough winding configurations to accommodate SuSy?

There is a priori no preferred number of holes. And whether the theory is SUSY or not, depends on various properties of the compactication space. If you consider 2 dimensional Riemann surfaces, then only the space with one hole, ie the torus, will give rise to a SUSY spectrum.
Quote by sneutron
Why is winding number considered "productive"?
A really dim guy might expect that a string 's' with winding number 2 would decay spontaneously to s + s (which is not seen in the real world?).

It primarily depends on whether such a decay is energetically favored. For wrappings on flat tori the mass of a string wrapping two times the _same_ cycle is indeed the same as two strings wrapping each one time. So one cannot really distinguish these, and calls this a "bound state at threshold". The situation is more interesting if there are several cycles. Then in general a string with wrappings (n1,n2,m1,m2) has less mass than two strings A,B whose wrappings add up like n1A+n1B=n1, etc. Thus one considers this string as bound state of those others. When the wrapping numbers are non coprime (ie, have a common factor), then the situation is again degenerate and one has a bound state at threshold.