Here is an alternative opinion.
Agreed that polyester does melt at
high heat, but if it is high enough to melt the fabric you would not have the baby anywhere nearby. The type of cloth (I think) you are refering to I call fleece fabric. The advantages with this fabric is that it is very easy to wash and dry quickly and it does not shrink or need ironing
This is good where babies are involved
I actually made a couple of baby sleeping bags for my daughter out of a double layer of fleece when she was a toddler (that way they can't roll out and get uncovered and wake up mum & dad hollaring because they are cold!) I do know that a number of other members have made baby quilts using minky/fleece happily - but off the top of my head I cannot remember who.
You mention that you haven't prewashed the cotton - have you checked for colour fastness? ie will any of the blighters bleed colour on the first wash? If they are all soft colours you are probably ok.
I would probably finish up making the top first. Then you have the choice of washing it to pre- shrink or you could just carry on and make it up and then when it does have it's first wash it will get the soft & puckery look, that happens and some people like especially for quilts that are meant to be loved & used hard, as opposed to show quilts.
No I don't think that you will need batting as well as fleece, unless it is destined for a particularly cold area.
Finishing - you could bind or you could just fold under the seam allowance of the cotton and stitch it down flat to the fleece. Fleece does not fray, if the blanket you are using is larger you could cut a fringe into the fleece if you wanted to.
My technique for just preshrinking fabric that is still in yardage (but will work fine for cuts) is lay it in the bottom of the bath tub with as few folds as possible, with the plug in. Boil the kettle and pour over, add a second kettleful of boiling water and rub over with a spatula or fish slice to make sure the fabric is completely soaked, adding hot tap water to make sure that it is fully immersed. Leave it for an hour or 2 to cool down/get side-tracked with something else. Drain the water out, squeeze out as much water
without screwing/twisting the fabric up and get it outside to peg on the line as flat as possible, use lots of pegs about every 6-8" This method avoids getting lots of creases in the fabric that then have to be got out with ironing. If it is small pieces lay them flat & square on a towel to dry. It doesn't necessarily need lots of agitation or soap to shrink, just hot water, in my experience.