Definitely driving 20 miles for $5 makes no sense. If you had several other shops already lined up in the same area, then adding on the $5 shop might make sense.
Companies vary a lot in their attitudes towards shoppers and the low pay jobs. Generally I have not found that taking the trash helped with getting treasures. When I took the trash I only got offered more of same. It is the marginally okay jobs, done well, that I have found makes a difference. As an Independent Contractor you are NEVER obligated to request or accept any job.
That being said, sometimes a scheduler will call with a shop the need to get filled. They may or may not be able to bonus the job enough to make it even marginable. Help them if you can, but know your companies' clients well enough to request other, better paying jobs specifically "I have really been hoping to do the ABC Client. Can I swap with you and do this job but also get an ABC?" Sometimes they can do it, sometimes they will state that another scheduler handles those. See what you can negotiate for yourself. This is a business you are trying to run, not a charity or a frienship service.