I've used Pellon 360 tear away stabilizer in the past, but I keep hearing such wonderful things about Sulky Solvy that I might try that next time. I usually never use poly thread but this is the one exception. I base my needle size on the recommendation for whatever thread I buy, but if there are multiple sizes recommended, I go with the smallest & buy a Microtex needle.
I make my applique as units. So I would apply all the fusible (windowing it) and then press between parchment paper. If an edge is going to be encapsulated inside the unit, I don't stitch it. Once I have my unit prepared, I peel it off the parchment paper, fuse it to my background, put the tear away stabilizer on underneath the background fabric & then get to stitching. I completely agree about going slow & usually with my machine I need to go back over it a second time so there aren't any gaps in my stitching. When I'm happy, I peel off the stabilizer & move on to the next one.
Note: don't apply the fusible too far in advance because it will stiffen & be more difficult to stitch through. If this happens, you can warm it up a bit with the iron but that only works so many times.
Personally, I prefer the "lighter" look of buttonhole stitching or sometimes even a cute decorative stitch. With those you don't need a stabilizer & don't need to go over it a second time. I also always use cotton thread with those. If I don't want it super noticeable, I'll pick an Aurifil 50/2 in a matching color with a 10/70 Microtex needle. If I want to feature the applique stitching, I'll pick an Aurifil 28/2 or sometimes even hand stitch with Aurifloss in a coordinating color. With the heavier weight threads, I use a 12/80 Microtex machine needle or stitch by hand with a John James Gold'n Glide Milliners size 10.
Good luck! Be sure to post photos when you're done. And welcome to the applique community. I'm sure once you try it you will be completely addicted like the rest of us.