I've had a plumber to my house twice to fix these issues respectively: kitchen sink stopped up, then bathtub stopped up, now dishwasher filled with dirty water. Dishwasher was the original problem (standing water in basin) long ago. Problems have been temporarily fixed by plumber until now. Dishwasher is filled with dirty water from disposal. I cleaned all the parts that will unscrew in dishwasher. Is this all related to my dishwasher? Could my dishwasher pump be bad causing all of these clogging problems? Plumbing lines were cleared with snake when plumber came out, but now it's happening again, and seems to be started when I run the disposal (which is new, but also happened with the old disposal, too).

Apparently you do not have an air gap as shown in the first picture.

You need a "high loop" as shown in the second picture. Just tie the dishwasher hose up to the underside of the counter top.