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Ok, just to be clear, the "gain" listed for this pre-amp is downstream to prevent signal loss from the antenna to RF input of TV, right? It does NOT increase the actual signal gain of the antenna itself, correct?
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You're on the right track. A preamp can overcome loss in the cable and splitters that are connected to the output of the amplifier. No amplifier can remove noise or distortion from a signal and in fact, all amplifiers add some noise and distortion to the signal. So, no amplifier can do the job of an antenna which is to recover enough signal of sufficient quality so that noise and distortion will not mask or 'bury' the signal.
Your math is correct but incomplete, or based on a misunderstanding. The signal at the output of the amplifier would be +12 dBm... I suspect that you are referring to the noise figure however. (-5 dBm would be an extremely powerful signal level and virtually all consumer grade amplifiers are incapable of delivering +12 dBm output.)
If you are looking at a signal with a predicted NM of -5 dB (not dBm), the noise figure of the amplifier would be subtracted from the signal NM. For an amplifier with a noise figure of 3 dB the math would be {NM(-5 dB - 3 dB = -8 dB)} which means that there would be 8 dB more noise than a perfect or near perfect receiver could work with. The amplifier makes the net NM worse, not better.
The solution is to use an antenna with enough gain to produce a signal with a positive NM. Antenna gain is better than amplifier gain. Antenna gain adds to the NM of a signal, amplifiers subtract from a signals' NM.
FWIW: Measurement in units of dBm are referenced to one Milli-Watt (1/1000 Watt) of power. Signal power is commonly expressed this way. Noise margin can be expressed in dB, with the zero level (reference point) being the minimum signal/noise that a perfect or very nearly perfect receiver could recover. Signal power and NM are two different measurements and the units used are unique to each.
There is a low level of electrical noise generated by every atom in the universe. When we are dealing with very weak signals, noise generated by random electron movement in the antenna itself become significant noise sources.