F'n power brakes: How do they work?

Here's how to check your brake booster. Run the engine for a couple minutes to build vacuum. Turn it off. Pump the brake pedal 3 or 4 times til it gets hard and hold pressure on the pedal. Now start the motor. A good booster will let the pedal sink slowly as soon as the engine runs, if nothing happens the booster is bad.

Of course first check to make sure vacuum is getting to the booster. Those check valves rarely give problems but check for a crack by the hose nipple, etc. A new rebuilt booster should come with a new check valve. There should be a ton of vacuum at the hose, like 22" hg whatever the hell vacuum is measured in.

You shouldn't bleed the brakes with the engine running, don't know why but no one says to do it like that.

I just went through a maddening time with brakes on my tow vehicle- which sounds awfully similar to what you're talking about.  In my case, the vehicle in question is a 3/4 ton Dodge W250 Cummuns TD pickup.  I ended up having to replace both rear wheel cylinders, a good chunk of the metal lines, all the flex hoses, and the master cylinder.  Even now, the brakes aren't what I'd call "Great" or even "Good" but they are definitely "Passable".  I was going near mad running around the vehicle bleeding it.  The brakes would seem to have good pedal and then I'd crank the truck up and the pedal would go right to the floor.

A couple of things to try; you can tell me "Mike, go to hell, I've already tried that!" but I'm just trying to think of everything I went through...

-On Chryslers, bleed the passenger rear wheel first, then driver rear, then passenger front, then driver front.  Makes sure whichever wheel you're bleeding is in the air and higher than the others, so the air goes towards it.

-Do you have ABS?  My truck has rear ABS, and there is a bleeder on the abs valve (located in the rear of the truck, in the frame rails).  Bleed that first.  Check that, bleed the Master cylinder first, (disconnect the two lines, buy two 12" long lines, screw them into the MC, bend them and dunk them into a vat of brake fluid, fill the MC, pump until no bubbles come out, reconnect the vehicle brake lines)

-Do your front flex hoses connect using a banjo fitting?  Did you replace the crush washers? 

-Are the rear brakes drum?  If yes, are they adjusted?  Drum brakes with too much shoe-drum clearance can cause a sinking feeling in the pedal. 

Basically what I've found is that a power booster can often exaggerate any problems with the rest of the brake system, so even if the brakes feel "OK" with no booster, they may be pretty marginal.  Or there may be a tiny leak, like you mention. 

I've never heard of bleeding with the engine on.  I have heard some Chrysler products require power bleeding, see a mechanic, etc, but I managed my truck just fine after going through everything above.  I plan on replacing the front calipers at some point soon, because the originals are 20+ years old, replacements are cheap, and  I want to install some towing-rated pads anyway.

I just checked, and it does in fact now stop properly from about 25mph (the best runup I can get on the neighborhood street and still have room to brake without running off into the farm at the end. If I started in the farm and did a runup towards the pavement, I could probably hit about 50 - but then I'd be pointing a potentially-not-stopping 4000lb missile at a neighbor's house) and the front brakes can totally hold the truck at WOT (and it does a wicked one-legged burnout - so says the absolutely trashed piece of shit spare tire I had on that corner).

I suppose this is consistent, since some parts of the brake system hadn't been replaced at last testing.

So the only real problem seems to be that there's no pedal feel whatsoever. In my opinion, this makes it 'safe' enough to pass inspection. Only problem is that inspection is done by Low-Rent Mechanics who are going to try to sell me brakes (pads and calipers, no doubt, completely ignoring the actual problem) at Hostage-Markup and get into an argument with me about "I can't let you drive away with that truck to fix it yourself"

Informationally, here's the butcher's bill on the brakes thus far:
- New RF brake hose (Originally missing)
- 2x rear wheel cylinders (originals leaked like crazy)
- RF brake hardline (original was cut. With dikes.)
- RR brake hardline (original had to be cut to remove wheel cylinder)
- LR brake hardline (original had to be cut to remove wheel cylinder)
- Pads, shoes all around

That is very interesting about the overboost problem, though. I'll have to keep that in mind if I wind up with some worthless asshat doing the inspection.

You mention"So the only real problem seems to be that there's no pedal feel whatsoever."

I forgot to mention I experienced the same thing, but carefully go over the rear brakes again.

Double-checked the rear today (and by double-checked I mean I took it all apart and put it back together again). That means it's definitely down to either 'designed by an idiot' or 'overboosted brakes' - I'm going to err on the side of not believing for a god damned minute that anyone involved in designing the 1st gen Dodge Dakota had the slightest idea what they were doing. If The Man doesn't like my brakes, I guess I'll cough up the $100 for a new booster (because god knows if I start digging in the junkyards I'll just end up with other problems)

Or maybe I'll figure out how to mount the MC straight to the firewall and pass it off as 'Power brakes? OPTIONAL EXTRA!' (and for extra authenticity, I'll delete the power steering pump)