Interacting classical strings? May classical strings be interacting?  
I would guess no, I can not see any way to break a classical closed string in two of them (the "pants" diagram); but maybe I'm missing something.  
 A: Yes, of course, classical strings are just the $\hbar\to 0$ limit of "ordinary strings" and they do interact although the rate goes to zero in the limit, too. The local picture of the interactions used for "quantum strings" should be interpreted literally and it does allow strings to interact even if they're "classical":

On the picture, you see the "type II" crossing over interaction which is the only one in the absence of open strings. Two strings intersect and they get split at the intersection points; the 4 endpoints get connected differently. This interaction is weighted by the amplitude $g_{\rm closed}$.
In the middle of the picture, you see the "type I" interactions which require open strings: two endpoints may either merge, or a string may split at any internal point. This interaction's amplitude is proportional to $g_{\rm open}\sim \sqrt{g_{\rm closed}}$.
These processes, as indicated on the picture, could rearrange the strings even if they were stretched to the astronomical size – if they were cosmic strings. Such interactions of large enough "cosmic" strings are actually essential for various models of string cosmology to work, e.g. Brandenberger-Vafa "string gas cosmology" and its newer versions.
A: 1) Usually, but not always, the word classical in physics refers to the limit $\hbar\to 0$. 
2) Perturbative string interactions of open and closed strings are governed by the open and the closed string coupling constants, respectively, which are independent of Planck constant $\hbar$.
