Firstly, my car is a 1998 E300TD with 240k miles on it. I got the car in 2007 with about 130k.

About 9 months ago I started experiencing what I am calling "Flat Tire Syndrome" (FTS). I don't know how else to describe this problem... It sounds and feels like I have flat tire, complete with the "whump, whump, whump" sound and the car shaking. Seems to happen more after I've been in stop-and-go traffic. I thought it was a tire going out of round as it heated up so I tried swapping wheels for the spare. I thought I confirmed my theory and got a new set of four. Sadly, the problem came back a couple weeks later.

Generally, stopping the car for a while fixes things. As I'm typing this I realize that I don't know if turning the car off may be part of the resolution. I'll have to try it next time it happens.

Most recently the FTS was accompanied by (or followed by - not sure which) a soft brake pedal. I got where I was going and an hour later everything was fine again. I think the soft brake pedal is a new component in this equation. The degree of softness also seems to vary.

Since I got the car in '07, I've had a pulsing when I step on the brake at certain speeds. Actually, it's more like "uncertain speeds" since it seems to vary some. I've replaced all four rotors and have pretty new pads on all four corners. I haven't seen any differential pad wear. Wheel bearings seem to be OK. Suspension and other components also seem to be OK. I don't feel a lot of shake in the wheel but I do feel some. If it's wheel related, I think it must be a rear wheel. If I had to guess, it would be the right rear wheel.

Is it possible these two issues are related? Is it possible that, even though there's no differential pad wear that I have a stuck caliper piston? Is it possible that the soft pedal is something to do with brake fluid?

Everything I can think of that would cause the FTS symptoms wouldn't be intermittent. Bearings, flex disc, warped rotor, warped rim, wheel balance, etc. would manifest a persistent shake. So I'm at a loss here and hopeful that you all have some ideas.

 

Everything is possible on the car and finding noisy part can be a nightmare.
Most of the time such problems are wheels.
I can't imagine brakes pulsing triggered by something different than bad rotor or bad wheel. Since you replaced the rotors, the conclusion is quite clear.
I test the wheels by lifting them slightly above the ground, putting small item next to the thread and observe the clearances while turning the wheel.
Than going under front suspension with set of big screwdrivers and pry bars might find bad bushing, or joint.

 

Mine has had a lot of that and is gradually getting better. For me it's taken
front ball joints
rear control arm bushings
rear bearings
front and rear rotors
front control arms and sway bar bushings

 

After my commute this morning, I'm almost certain it's brake related. Pedal got squishy when the shake and noise started. Then, at low speed, there was a rhythmic metallic scraping instead of the whump, whump noise. There's also visible evidence of overheating - the sides of the left side rear pads have turned white like partially burnt charcoal.

I think the left rear brake is overheating and the rotor is warping. That must mean a stuck caliper, right? How do I check caliper function?