Forgive the long post. I have some work experience in detection of aircraft by radar and feel like talking.
Radar works by detecting radio-frequency (RF) energy. General types are active radars, which transmit RF energy, and passive, which do not, using environmental sources instead. Active radars are either monostatic, where the transmitter and receiver are co-located, bistatic, where the transmitter and receiver are separated, or multistatic, which has one transmitter and many, dispersed receivers. Whatever the type, the energy from the aircraft must be distinguishable (by intesity, location, and/or frequency-shift due to motion) from all the other energy received by the radar. Most operational radars are active and monostatic.
To avoid detection by a radar, you must minimize the amount of energy it receives. There are many ways to do this:[ol]
[li]Reflect the energy away from the receiver. Thus the sloped shapes and smooth surfaces of stealthy aircraft. Even the pilot’s head/helmet is considered. []Absorb the energy, by wise choices of materials. []Avoid the RF energy by using terrain and the curvature of the earth. Fly low, and behind mountains. []Increase the clutter by releasing chaff or decoys. []Confusing the receiver with energy (jamming). One aircraft, at a safer distance, will transmit energy at the radar to blind it or confuse its signal processing, preventing the detection of other aircraft. [*]Stop the radar from transmitting and/or receiving. Often by either destroying it or scaring them enough to not turn it on. [/ol][/li]
Radar cross section (RCS) is a measure of how much energy is returned from any given look angle. A large-RCS aircraft is easy to detect, while a low-RCS aircraft is much more difficult. Designers of low-RCS aircraft must make engineer trade-offs about where to reflect the energy. You cannot minimize reflections from all angles. Typical stealthy designs minimize the reflections for, in priority, the front, front-low, front-high, rear, rear-low, rear-high, sides, beneath, above.
On specific questions/points:
A RADAR wave hitting the underside of the bomber as it flew over an installation would have a large area to hit, surely?
Yes, aircraft usually have large a RCS from beneath. But aircraft do not fly directly above hostile radars (unless attacking them), and surveillance radars typically point out, not up. It’s more important to avoid detection while approaching and leaving, than while above it. The targeting radars of surface to air missiles (SAMs) can sometimes detect aircraft above them, but no one flies directly above a SAM (more than once 
).
I heard that a special paint was developed that absorbed RADAR waves, yet, if this is true, why not paint more planes than the B2 with it?
The paint is expensive to maintain and has some environmental restrictions. The surface shape and the material underneath the paint are more important. No paint would make an F-16 stealthy.
Rumors are that the Russians have developed better radar to detect “stealth” planes… beggining of a stealth tech race?
Stealth can be negated several ways. A radar with a powerful transmitter, sensitive receiver, sophisticated signal processing, and proper doctrine can potentially detect a lot. Bistatic radars can circumvent stealth by detecting the energy that’s reflected away from the transmitter. Passive radars have a similar advantage. However, these anti-stealth technologies will not make stealthy aircraft obsolete. It’s always better to be a smaller target. And there’s other options for mitigating radars (see points 3-6). So, yes, there’s tech race, as always.
I thought it was the opposite? My understanding was that radio waves will reflect off flat panels, and very little of it gets back towards the source. (Unless the surface is perpendicular to the radio wave, which is unlikely.) Complex curves (like on conventional aircraft) will scatter radio waves, and some of it gets scattered back towards the source, where the detection system is.
In radar terminology, “scatter” is the preferred term for any reflection. “Reflection”, and especially “specular reflection” (which no one has used, but I think is what’s in mind here), is more a term for the optical limit (i.e., short wavelengths). Just use “scatter”.
On preview: Lemur866 is exactly right. Stealthiness is a continuum, not an either-or. In the industry, we use the term “low-observable” not “stealth”.
