I love how the gray setting fabric sets off your charms. I think you'll be surprised at how nice that will look when it's quilted and washed. The advice that "It will quilt out" is quite true of most little imperfections in quilts. Batting that is all or mostly cotton will shrink a bit when the quilt is washed, guaranteeing a certain amount of puffiness, and it seems to me that's the goal in making a quilt for warmth and comfort as well as eye appeal.
A lot of people advise against steam because it can cause some distortion, but on the other hand it's used by Eleanor Burns and quite a lot of accomplished quilters. It's probably more important not to pull and tug at the fabric when you use the iron and to lift and press the iron rather than shoving it around. It's also important when sandwiching the quilt to lay it firmly enough to be straight and neat, but to not yank it in a way that might distort it. All that sort of thing you will gradually get a feel for as you go along. I have a hunch that the weight of the top as you hung it on the vertical design wall may be contributing to the unevenness that I'm seeing and that it may not do that when it's horizontal - perhaps on the floor, if you can't find a table large enough to work on.
What style of quilting are you planning to do? If you have a walking foot, a simple wavy design back and forth starting in the middle of one side and working out to the sides (switching direction for each pass) would probably be the easiest approach. If you want to try free motion, there's a lot of help on youtube, such as this video by Patsy Thompson:
https://www.youtube.com/watch?v=39I5A3iyCtw. Probably you want to get something like an old sheet (with the same kind of batting you plan to use) and create some (lots!) small practice sandwiches before you tackle the real quilt.
I hope you'll show us the quilt when it's done. I'm sure it will be delightful.