What do you recommend to clean the exterior trim around the tires? Regular car wash soap does not d...

Try any of the "blackening" products that do not contain silicon, they are made for restoring faded trim to its like-new condition but may work in this case.

Give autoglym bumper care a try.  It worked really well on restoring the bumpers on my old Citroen BX.  I also use it on current cars on all rubber/plastic areas.  It can be greasy but creates a great finish.