Let's start with..
How would the bomb explode underground, in the absence of any oxygen?
Bombs don't require oxygen to explode, they just need an explosive material and some form of detonator. Fire requires oxygen to continue burning but explosive materials do not.
An explosive material, also called an explosive, is a reactive substance that contains a great amount of potential energy that can produce an explosion if released suddenly, usually accompanied by the production of light, heat, sound, and pressure.
Explosive materials may be categorized by the speed at which they expand. Materials that detonate (the front of the chemical reaction moves faster through the material than the speed of sound) are said to be "high explosives" and materials that deflagrate are said to be "low explosives". Explosives may also be categorized by their sensitivity. Sensitive materials that can be initiated by a relatively small amount of heat or pressure are primary explosives and materials that are relatively insensitive are secondary or tertiary explosives.
How can a bomb that can sometimes be scratched/damaged from pressure, in a creature's belly, underground, be signalled with an overground detonator?
There's a certain amount of "plot magic" here but the bombs aren't really under any pressure underground since they have been swallowed whole.
As for the radio signals, it's true that these don't propogate very far underground, usually just a few feet but, in this case, that's exactly how deep the graboids are. You can tell since the explosions make it to the surface with relatively small amounts of explosive per device