Many new machines have added additional drains on the door gasket to help eliminate the issue. I can tell you the new GE ultra fresh washers have added a micro ban material that mold mildew and bacteria will not adhere to. That is my #1 choice if worried about the issue.

Ever yes it can, I personally have a white quartz counter and it is fantastic and has yet to stain from wine spills to oil I am very satisfied with its durability. I say yes as over time anything is possible with some type of product or chemical that can permeate it.  I would give it a thumbs up as a counter material.

First check the filter at the bottom of the unit to make sure nothing is blocking the drain if not you will need to disconnect the drain hose from under the sink and lower it below the unit see if water comes out that will help eliminate if the drain is clogged or the sink line.