you should be able to unplug the temp sensor and I dont think it would give an error, the car doesn't really car what temperature it runs at, even if you overheat it you will only get the red light on the dash, no warning light I believe. certainly true of CF2, not tried on CF3 although I may give it a go later for a giggle now.
that means you probably have two seperate unrelated issues.
things to check [for engine temp]
run your fingers down back of the rad and see if the fins on the rad are soft and or crumbling, if they are, the rad is shot and needs replacing. if the rad needs replacing the thermostat will want replacing with it.
you could replace just the stat but that will show up the weak radiator and they tend to mask each others failing and typically only live around 5yrs.
you can also check the cars ECU via the software as the temp the gauge shows and the temp the ECU sees are from different sensor heads, so gauge could be showing different temp to what the ECU sees - worth checking. in the unlikely event that they show drastically different temps change sensor - else investigate rad and stat.
then you can check the ECU for fault codes whilst you were there as you probs have something random going on.
also note that in this cold weather turn your aircon off as this pulses the fan for the AC system and will slow your cars warming up!
1999 156 1.8TS Sport Pack 3
2001 166 3.0 V6 24V Super
2003 156 GTA 3.2 V6 24V
2005 156 1.9 JTDM-Jet Sport
2008 159 1.9 JTDM Lusso
1997 GTV 2.0TS Lusso
2001 147 1.6TS Lusso