Hello
I say, I think you wrote down the problem areas in nice descriptions.

A bad ignition system shows itself, especially, if the wires on damp or wet days. This system is the very first suspect.
The four wires are dependent on the one and only wire from the coil, since it feeds four more. Check it to a good ground point.

Small amounts of missing is usually narrowed down to one bad spark plug wire.
A hard start or no start can be either the rotor button or distributor cap.

These two items below are related to an empty fuel rail or loss of rest pressure during off cycles.
Pumps not running when needed, a fuel system relay. A leaky FPR diaphragm can equal flooding gas into the intake manifold via the vacuum line, or check ball on the return side line side to the fuel tank can happen. Low fuel rail, maybe?

You wrote;
"First, it will struggle to fire up, and then it will try to stall if I go anywhere near the throttle."
Sounds very lean here?

It takes time to fill the rail up when the engine needs more fuel because it's colder.
A low fuel amount and/or pressure will not make a fully atomized spray cone. All of this makes for a really rough start and running moment.

You wrote;
"It may also be worth noting that this issue seems to be exacerbated when the amount of fuel in the tank is low. After it is started several times and has awhile to run, the engine seems to run normally."

The in-tank fuel respond this way to the engine when it's not working. It is needed at this more crucial time, to work correctly, when the tank in below half full.
Just as bad if the tank is near empty like when the car is needed to climb a up steep hill is.

It's another symptom clue "if" it relates to a "time of" or "when" occurrence.

Personally, I think you are paying attention to your car just fine!
It's talking and you are listening!

Now, think about a common thread defect that will lead yourself to the biggest possible gremlin to dig for!
I could tell you what I would do, but what kind of entertainment would that be and do for all of us?
(:-)

Phil