Is it common for a cold reciprocating engine to stop at idle?
What you experienced is common, but it would serve you well to better understand your engine.

There are quite a few engine options for the PA28, so without more specifics I cannot give you specific guidance, but I will address this in a general form.

Spark plugs foul from several different sources.  The leading sources are from excessive unburnt fuel, and from oil.  The unburnt fuel can cause carbon and fuel based buildup on the plugs, or it can come in the form of lead fouling, where lead deposits bridge the spark gap or insulator in the spark plug. Oil fouling tends to happen in higher time engines, but can happen in new engines during their break-in period.  Normally, running at cruise power keeps the plugs clean by scavenging these deposits off.

Again, engines vary, and manufacturer's recommendations vary, but in general, I run with the engine lean during taxi. This reduces buildups and will clear some fouling.  Because the engine running is impacted by engine temperature, I generally do not do a run-up until after taxi, when the engine is warmer.

If an engine is running rough, or the mag check fails, or the engine repeatedly stalls, I will do a clearing run-up, where the engine is run at a higher power setting, typically 1800 to 2000RPM and leaned until it will just continue running. This creates higher temperatures, and tends to burn off oil or other fouling.  This is done for anywhere from 1 minute to about 3 minutes, and then the mixture is enrichened and a normal run-up and mag check are done.

Magnetos do not foul, rather spark plugs foul, and typically the R mag will run the top plugs on one side of the engine, and the L mag will run the top plugs on the other side. Bottom plugs tend to fould more, especially if there is oil fouling. But the point is that whether the L or R mag fails, or runs rough, doesn't tell you exactly what spark plugs it is. A cylinder temp gauge or other instrumentation may, but usually it is determined by disassembling and inspecting the spark plugs.

So to get to the OP question: Large drops at runup are a sign of either a condition which may resolve itself, or conditions which are causing the engine to run less effectively. If they can't be respoved the plane needs to be seen by a mechanic, or someone more familiar with it.  If the mag is rough, and the engine has just been started, then it is premature to perform a normal mag check.  Let the engine ward up more, during taxi, and then check the mags.  The engine should not die at idle.  If that happens repeatedly after the engine is warmer (not even up to operating temps), then there is a problem, and the engine should be checked out by a mechanic or someone with more experience.

In addition to not clue as to what engine you are dealing with, we also don't know the local meteorological conditions. If you are having problems at 0F it is more typical than having problems at 35F.  But in general, engines should not stall at idle even when cold at 40F temperatures.

A final comment. I have trained many CFIs, and I spend quite a bit of time on engines and their operation and work-arounds.  Much of this is not in FAA publications, or even training publications. I find that many CFIs are not a knowledgeable about engine operations and related methods. Being engine smart will help you in your flying, and it will reduce your risk.

Addendum: 

As a general rule:  Lycoming: R Mag top left plugs, cyl 2, 4 and bottom right plugs, cyl 1, 3.
Continentals: R Mag fires the top plugs, L Mag fires the bottom. The impulse is the R Mag, as the top plugs tend to get less oil fouling.