The quick answer is -- yes, it will explode.

Nearly all explosives carry both reaction components (or an unstable single component like TNT). Effectively, all they need to react is a warm enough temperature and a suitable trigger. On the other hand, without an atmosphere to conduct sound and shock waves, an explosion is silent until the actual debris,ejected from the explosion hit. Thus, the characteristics of damage are substantially different.

It is also true that regular bullets will fire from cartridges in space -- Apollo astronauts used a "thumper" which fired such shells to act a sources for seismic studies of the Moon...Since there is no air to slow or deflect it, a bullet in space hits just as hard no matter how far it travels, (unless you are in a gravity well-- then it picks up a lot of additional energy).

This is a very good question. The shock waves from a bomb (which cause the damage) require a medium to propagate. Bombs are much more dangerous under water than they are in air, for example. So in the absence of a medium i.e. in outer space, one would expect less damage. However, bombs/explosive are associated with rapid chemical reactions which create a lot of gas and the rapid spreading of these gases should enable bombs to cause some damage even in space.