And you are of course correct on the term system temperature as being an all inclusive term for system noise, varying with the ambient air and component temperatures.
Since ambient air and component temperatures vary with the weather and component cooling methods I usually consider the 1st amplification stage's NF as setting the noise floor.
Definitely the right way to think about how it works. The beginning parts are much more important than the later stages.
This is also the reason we recommend a good antenna and cabling. The biggest increase in performance are in the beginning because once the signal is messed up it can't be fixed in the later stages.
On the filter, I have the understanding that it reduces both the signal and noise ~2db inside it's pass band but about 40 db outside.
So if there were no LNA in front of the filter, just say a ~2dBi gain antenna, the gain of the antenna would be completely eliminated by the insertion loss of the filter.
OTOH if an LNA with a 0.23 dB NF and ~18dB gain were put between the antenna and filter the antenna gain would only be reduced to ~1.77 dB and the 18 dB of gain would replace the losses due to components temperatures downstream of the LNA, thus the S/N ratio would remain the same downstream of the LNA in the pass band of interest.
Gain numbers don't quite work like that. dB are relative measurements so you have to know what you are comparing it too. Also dB is logarithmic scale so every 3dB change is about half or twice the power. Something with +3dB higher means it has twice the power. Something -3dB lower means it has half the power. Something with +6dB means it has 4x the power. Something with +9dB means it has 8x the power.
The dBi (it ends with the letter i) is a relative measure compared to a isotropic antenna. A 2dBi antenna means that the antenna has 2dB more gain than an isotropic antenna in the main direction. The other way to think of it is that the antenna instead of antenna receiving energy in all direction it will instead get 2dB more in some directions and a few dB less in other directions.
A dBm (it ends with the letter m) is the power relative to a milliWatt of power. So on an absolute measurement the prostick can measure power levels down to about -90dBm. This is why receivers are usually speced in the lowest power level it can detect in dBm.
You should think of it more like this: A plane has a 200W transmitter which sends the signal through the air and it is picked up by the antenna. The antenna turns the signal into a voltage which is in the microV range. This microV signal then has to be transmitted through the cables , amplified and filtered and you can estimate the final signal based on the gain of each part. So your microV signal has a -3dB loss from the cable, then a 18dB gain from the amp, then a -2dB loss from the filter, and finally a +50dB gain internal to the receiver chip (controlled with the gain setting in piaware). With the right oscilloscope you can measure the voltage at the different stages.
We output the final gain on the local web interface as a dBFS measurement (dB Full Scale). This is a dB measurement based on the receiver scale and it isn't calibrated so you don't know the actual dBm measurement without an oscilloscope. You can see this measurement by clicking on a plane on your PiAware map and then going to the section on the right and read off the dBFS measurement.
Also, there are *lots* of other losses than what we have listed. The main thing is that you get a good signal and then not mess it up in any of the stages.