Boy did you open a ball of wax! LOL. That being said I will do my best to answer some of your questions from my experience in both entering shows and judging them.
Quilt shows and the judges vary widely in that some judges are certified (by different institutions like the National Quilting Association) and some are not. In the majority of shows judges are volunteers.
There are specific guidelines that should be followed, some of which include....entered in right category, rules followed (i.e., sleeve attached), squaring of quilts, points matched, uniform stitching, color choices, pattern choices, nice corners, good thread tension, etc.
And yes, a tape measure is routinely used. Not to measure "all" lines but in cases where eyballing doesn't look quite right in an area they measure to make sure. A good example of this is when one side of the binding appears to be larger or smaller than another.
Quilt judges are not supposed to be biased in their choices but they, too, are human. Some judges prefer contemporary, while others are tradidtional die-hards and do not like anything "new".
Most of the larger quilt shows, i.e., state fair competitions, will tell you who the judges are. Sometimes you can contact the entering organization and find out ahead of time. Trying to get a volunteer judge is often hard and they do not know ahead of time who that judge will be.
All this being said, here are some guidelines to follow when entering a quilt for judging: Make sure you have entered quilt in right category and have followed all requirements, i.e., attached sleeves. Entry should be clean and free of all odors. Entry should not be from a kit - most shows specifically disallow this. Color choices and pattern choices are important. Blocks/sashing should be uniform, quilts should be squared. For best results, corners should be mitered. Quilting stitches should be uniform with good tension. And one of the most important rules that judges critique is that the batting should fully fill the binding -- this means if you are sewing a 1/4" seam your batting should extend about 1/8" more so that when it is completely attached you cannot pinch the edges and not encounter batting.
I hope this helps and good luck!