Why does the house feel toasty when it’s very cold outside, but cold when it’s only a little chilly out

I'm going to suspect poorly insulated ducts running in unconditioned space as a first guess. I don't know the layout of your system, but it's very common to take duct runs up into the attic, and even if they are wrapped with duct wrap (R3-5 typically) that's abysmal. If the ducts are poorly wrapped or not wrapped in spots, it gets worse. R-50+ is far more appropriate for hot air in a cold attic.

As for the furnace exhaust, it should be fairly warm - otherise you get to visit the series of questions that are on here about furnace exhaust (high-efficiency condensing) freezing up in low temperatures and shutting the furnace down.

Fundamentally, if your furnace is running 24/7 and the temperature continues to drop, it's undersized for the load - but one of the best ways to deal with that is to reduce its load by improving insulation and air-sealing (caulking things - the most boring, most bang-for-the-buck way to improve efficiency in most houses that have not already had that done). You could also add a second furnace to take some of the load, but then you'll be burning even more gas. You do not want a furnace that is much larger than your "design load" (ideally, your furnace runs 24/7 and maintains temperature on your coldest days) since an oversized furnace is inefficient .vs. one that's just the right size.

Just a guess, but the colder it gets the longer the boiler cycles and the hotter the delivery water temperature gets so therefore the radiators get hotter before the thermostat shuts the boiler off.

In moderate temperatures, the thermostat shuts the boiler off before it gets very warm and so your radiators are barely above room temperature.

Check to see if your boiler has any kind of temperature control on the delivery water going out into the system. You may be able to adjust that for better performance in moderate weather.

Good luck with your project!