Welcome from NJ. As others have said, your goal is evenly spaced stitches - not necessarily small. Your stitches will get smaller over time with lots more practice.
I, too, use 'between' needles for hand quilting. As well as specifically-labeled hand quilting thread. I use a metal thimble from Thimblelady. It is an open-ended thimble with nice dimples to hold the needle but you push with the pad of your finger vs the top. I don't use a protector for my under finger - just build up a callous, lol.
As to your cotton batting...I use 100% cotton batting (Warm & Natural) all the time. Many people find it hard to hand quilt through. I guess I've just never known any better/differently so I don't really have a problem with it. I do find using a 'thread conditioner' to help tremendously - not only with making it easier to pull the thread through the quilt sandwich but also to help reduce thread tangling. I use Thread Heaven. I believe it may have been discontinued or rebranded under another name but it can still be purchased but you might have to chase around some for it.
Wool and silk are also other great natural battings. Silk batting is very pricey so unless making an 'heirloom' quilt I would not spend the $$. But it does have a heavenly 'hand' to it.
Good luck with your projects and enjoy the process.