Batch Process 4,000 PNG Files

1. The edges on some of these PNGs are pixelated and when I create the stroke, it's jagged. I need to clean this up before creating the stroke. 

It would be helpful to provide a layered PSD of the files with the stroke, are you using layer styles/fx? What if it's possible to have a clean stroke without cleaning up the PNG edge, would that work?

2. Also, some of these images have empty space in the middle of them. I need to fill those with the white background as well. 

I don't believe that the samples provided so far show any example of this?

3. Finally, I need to resize the emoji to fit the artboard. (These are NOT for print. I already have print files.)

How big is the artboard in pixels compared to the emoji size in pixels (do you perhaps mean canvas, not artboard)? Have you considered working from the print files? The icon is only 160px in size, but the PSD is 2000px. You really don't want to scale up the images!

This mostly sounds like something that an Action can do, and then the Batch command or Image Processor or Image Processor Pro etc. A custom script might be required, but that isn't known yet.

Point #2 might require special handling, so seeing a small range of different images that need the interior filled with white would be helpful.

You either design a process that works for all, or you will need to isolate the exceptions and process them differently.