ChasKSC

I found putting the soap on the wash mitt your using is best. About half way thru I'll hit the mitt while it's in the bucket and it suds right back up. I believe there is a Adams video showing to put the car shampoo onto the mitt not in the bucket then fill with water.

When I lived up north, I did a very similar method and through multiple trial and error sessions (more errors than anything else), I found the best solution for me.   I would put the shampoo in a bottle about the size of a Mason Jar.   Put a few ounces of shampoo in the bottle, fill the bottle about 3/4 of way up with water and head to the car wash.  Once you get everything ready, shake the jar to mix the shampoo and water, then pour it into the bucket.
 
The mixed shampoo with spread throughout the water vs squirting it directly into the bucket which causes about 70% of it to sink to the bottom of the bucket. The goal is to disperse the shampoo as widely as possible in the bucket. If that doesn't work well enough, then stir it up before adding additional water.

I think Chris and Chris answered the questions.  As for the brakes, that is not a problem that I have experienced with vehicles with disk brakes.  However, if you have drum brakes, I have experienced that multiple times.  The solution is to drive it a little bit and it goes away.

There should be (could be?) some drain holes that help get the water out of those types of areas. If there are some in that area, make sure they are not clogged. 

If I understand what you're experiencing, you can use the pressure washer to spray the suds and water out of the cracks, then use a Sidekick or another dryer to dry. But beware, there's a trick to using it 🤦🏻‍♂️😂