I agree with pressing it from the reverse with a towel underneath. I would not advise washing the top before quilting, though. I would be afraid that the background fabric would shrink and distort. Once fabric is quilted rather thoroughly to batting and backing, the batting controls shrinkage. In other words, at that point the top fabric would not be able to shrink more than the batting shrinks. Fabrics that are washed on their own (without quilting) can shrink and distort terribly, especially with the cross-stitch already there (which will affect how the portion of the fabric underneath the embroidery shrinks), and especially since you have no way of knowing what that fabric is. If it is cheaper fabric, which I suspect is used in many kits, you really do not know how it may shrink. I would definitely not take that chance.
Which bring us to batting. I think the end result would be better if you use an actual batting rather than flannel. For easy quilting, I suggest Quilter's Dream request weight or select weight. This would be similar to flannel but would provide more opacity (opaqueness?) so backing would not shadow through to the white top. Especially since you are hand quilting, you want at least a little bit of loft to show off your hand quilting stitches.
Flannel for the backing would be very nice. However, I would wash and dry the flannel twice first, as flannel has a tendency to shrink a *lot*. This is not a problem for machine-quilted quilts with moderate quilting, but it could end up distorting a hand quilted quilt if it is not thoroughly prewashed.
The brown spot will come out later; it is not necessary to remove it before quilting. If necessary you can use RetroClean to get the spot out (
http://www.retroclean.com ).
The broken lines most likely mean that the embroidered blocks were intended to be cut apart and then sewn back together with sashing between the blocks. You do not have to do this if you do not want to. If you do not cut the blocks apart and add sashing, then I would hand quilt over those lines.
The blue markings may or may not come out later depending on how old the top is and how it was stored. If the marks do not come out, it simply conveys the message that this is a vintage style top and adds to the charm. I would not worry about that.
I would worry, however, about the colorfastness of the threads used. Many embroidery threads (especially reds) will bleed when exposed to water. This is another reason why I would not prewash the top before quilting. Once quilting and binding are finished, I would wash the entire quilt in a machine that uses a *lot* of water -- for example, the largest front loader at the laundromat -- and use Synthrapol for the first wash. You need lots of water in order to dilute any dye bleeds, and Synthrapol works to keep loose dye particles suspended in water so they can be rinsed away instead of settling into the fabric.