I was a TPMS developer for 5 years (2006-2011): lots of discussions and meeting. I need to write a .pdf to comment everything here.
All the system I developed could manage 4x2=8 Tires in memory so no problem for winter/summer change. If you have a TPMS Reset funtion you HAVE to reset the system everytime you mess around the tires (refill or change).
Hacking a TPMS at radio level is not too complicated: there were 3 main providers for the market: BERU, Schrader and Continental. Please do not do that, there are some collateral effect you don´t want, moreover there are much more easy ways to attack the beast as already commented.
All of them 3 main providers have propetary protocol and different working principles. At the end my TPMS adventure they were talking about a standard protocol compatible for everybody. After countless meeting and discussion I saw no agreement and I was pissed because lots of nice feature were going in the bin to make everything compatible back and forth.
The golden rule:
PV=nRT
this works almost perfectly in a air-tire (or the N2 bullshit low leakage gas) system. For runflat consider V almost constant, for non runflat there is a V change with P changing but not too dramatical (al long P remains in a certain range).
Anyway if T goes down P goes down and viceversa. If you ride after 15min the tires are warmed up generally (evet up to +20C from initial condition). So do not relfill your tire when hot, or if you do put something more to compensate.
Normally we are talking about 0,1Bar/10C. Please do your calculation regarding your local condition. Please remember the suggested BAR/PSI are for COLD Tires (not driven).
When the light is triggered depends on federal country law and system implementation. Normally there is a trigger when P-25% of reset value (if the car has a TPMS reset function) AND there is a low limit generally at 2,7BAR absolute (=generally +1Bar of ATM pressure, so 1,7 Bar when you read it with a normal manometer at the gas station).
Moreover what the manometer (tire gauge) says at the gas station is just the difference between the atmosferic pressure and the pressure inside the tire. The sensor has no clue what is the outside atmosferic pressure. We had issues when customers were filling the tires in Andorra (I was told cheap gas there and busy gas stations) at relative high altitude and then going down the mountains the tire were virtually loosing pressure...
Another golden tip: DIFFUSION. There are no perfect tires, the leakage rate is normally 1Bar/year but is depends if the tires are the giving traction or not (i.e. in rear traction cars the diffusion rate was 1,1 Bar/year and in the front 1Bar/year). By knowing the T and P at the reset condition and the T and P in the tires it is easy to calculate how much pressure was lost due to diffusion or low leakage AND the P/T gradient.
The best TPMS system I know?