I asked myself the same question several time, and frankly the answers I see here would have only partially satisfied my curiosity (topomorto's one being the one that gets closer to that), so I will try to add my two cents and make it more and more complex at each stem
1 ] Generally speaking, and with some degree of approximation, yes: sound waves "merge" into a more complex sound wave and nothing special happens, as the resulting wave is nothing but the sum of the two original waves.
This is a phenomenon (or better, a property of several phenomena) called linearity. When specifically discussing about waves one refers to the Superposition Principle, which is the basis of stuff like constructive/destructive interference:
Since sound waves are usually not boring sine waves, these illustrations should just give an idea
[edit: meaning that it's very unlikely to have such simplified A+B = 2A and A+B = 0 cases with two different complex soundwaves...]
2 ] ...but please always remember that sound is waves moving in a three-dimensional space, so to visualize a bit better what's happening take this nice drawing here
and add one further dimension: imagine spheres instead of circles.
So yes, two sounds merge, but they do so differently in different points of space - which is, amongst the other things, why you are able to distinguish which instrument is playing on your left and which on your right, as the way in which the waves will sum at your left and right ear will be different..
3 ] ..and furthermore, all this is definitely nice, although in real-real-world linearity is an extremely rare property, and only holds for certain cases. Luckily, when talking about sound, "most" of what happens is very well described in terms of linearity, but just as food for thought one should mention that there are subtle, minimal differences between reality and the physical model.
Other answers added extra phenomena to the game, but I think that goes beyond the question. Speaking about the sound waves generated by two (or, well, more) instruments, this should pretty much sum it!
edit: I see now that you also asked about what happens if sounds are not generated at the same time. Well, it doesn't really matter so much, the principle is the same. If the two fishermen in the above picture had cast their baits at different times, then the earlier one would have created waves that would have travelled farther (a bigger outside circle) and met the other waves in differenet points. The underlying principle remains the same!