basic questions

• "The idea is that if I then take my brush tool, having selected "Paint fills," then i can draw a big line over the square. Anything that crosses over the square will vanish leaving the lines on the outside of it, and it will also break down the outline of the square. However when I draw anything, it just disappears. Why?"

Most likely because you're drawing with Object Drawing Mode turned on. If so, Paint Fills will not work, turn it off in the properties panel or the toolbar (the icon right above brush mode) then try again. From Animate 21.5 onwards Paint Fills doesn't paint on empty areas anymore and will work in Object Mode as well: https://youtu.be/iY78Fl633Zo

• "When trying to add a gradient, I open the "color" tab and select "linear" or "radial" gradient. However when I use the eyedropper tool, the color selector goes back to "solid" color as opposed to "gradient"."

I can't see this issue on my end. It could be that you have Lock-Fill turned on, which applies gradients to shapes relative to their position on stage, so depending on where you're drawing in your screen you might not notice the gradient effect. This can also be turned off from the properties or the toolbar.

• "When arranging overlapping shape order all the "bring to front" "send to back" etc options are greyed out. They are not on locked layers, they are not on seperate layers. The shapes themselves are not locked. How do I arrange shapes easily?"

Sounds like another Object Drawing Mode issue. Drawing objects will always appear on top of raw shapes, you can change the order between drawing objects and groups but not raw shapes since they're "flattened" and so will always appear on the back. Here's a brief explanation on what object drawing does and when to use it: https://youtu.be/NgLPQpmBiNQ

"Can anyone explain why they would prefer this program to something else like photoshop of illustrator?"

For people who are well-acquainted and comfortable with the program these tools are more versatile and faster to produce assets than the programs you mention.

For production it is best to produce all vector artwork in Flash/Animate and only use external programs for raster backgrounds. Creating structures suitable for animation as you you design/produce the artwork is incomparable to convoluted workflows that require importation and constant external editing.

if you're just using animate to create drawing assets, start with an as3 project, not an html5/canvas project.  then let us know what doesn't work for you.