Hi Fan and welcome to the forum.
1. How much effect do warmer days have on draft? As I said, it seems my draft was fine on the 30-degree days and then got worse as the temp warmed up.
The chimney will tend to draft less as the outside temperature rises. The colder it is outside-the more that hot combustion from the stove will want to exit the chimney. If my stove is not burning and it's cold at night and then gets warm during the day my chimney will reverse draft once the outside temperature rises above the temperature inside the chimney.
2. How important is it to have a cap on the flue(s) when burning coal? Someone told me somewhere that coal fires definitely need the cap.
The cap is useful in helping to prevent a draft-reversal caused by a gust of wind. It also prevents rainwater from entering the chimney as well as any errant squirrels and birds. It is not a requirement for coal burning.
3. Could leaves be causing a lack of draft? Or would it be some other problem? Is it an adjustment needed on the barometric draft control?
I hope you cleaned the chimney or at least inspected it before lighting the stove. I would doubt it's leaves, there would have to be a ton of them in there to cause a problem. The baro damper needs to be set to limit max. draft of the stove. I run my Mark I at -.05" WC. You need a manometer to check this, there are threads on the forum about that.
Remember the warmer the chimney-the more it will draft. Before lighting a coal fire in the Mark II you should warm up the chimney by burning a small wood fire in the stove. Once everything is warmed up start to add the coal slowly. When lighting my stove I use charcoal soaked with charcoal lighter fluid, I cover the baro with foil and leave the ash door open until the chimney is warmed up.
Read here:How to Light a Hand Fired Coal Stove
Hope this helps out.