You can wash a quilt top prior to quilting, but it's a huge pain for very little gain. Hair washes off, but is still in the water. Some of the hair might come off and not be re-deposited on the quilt top, but some might be. It will help with the dander, though. The dryer is less of a problem because of the filter. Ever accidentally washed a pair of jeans with a tissue in the pocket? Most of the tissue winds up in the filter.
I agree 100% with SVAL that it is a big no-no to wash someone elses quilt top before quilting it. God forbid if it came apart during washing or bleed all over the quilt. Plus, you have to press them when they come out of the dryer.
Sometimes I have asked the top maker if could I add a border to a charity quilt to make it a little bigger - almost all of my quilts go to kids in foster care - and that makes them last a little longer as the children grow up. I have 1/4 inch seamed around the edges if there are a lot of edge seams, too, because it prevents problems while quilting.