(When we do a load of laundry, the washer will drain into the sump pump's tank, the tank will fill to a certain level, and then it will discharge all the water from that tank to the main line that leads outside to the city sewer.)

Rent is usually allowed as necessary living expenses, if not all rents will probably go to escrow to pay you later, if not you can toss them out upon discharge, I doubt you'll have a problem

I am ready to discharge them and fine someone new, but I have paid out of pocket expenses for inspection, appraisal, insurance and a HUD consultant that I am worried I cannot recoup or these will not be accepted by a new lender.  

Since it is a mixture, they are not able to drain it, so I have to get a hold of another company to give me an estimate.Thats no big deal oil and water will separate , you pump off the bottom using a clear hose as the discharge hose , once you see red you are into the good oil .

There are generally waiting periods of at least two years from the discharge of the bankruptcy.

@Matt Rasich When washing machines drain, they discharge a lot of water at high speed all at once and since you have no problem when you run the water yourself directly into the sink, then your problem might be poor venting which does not allow the water from the washing machine to drain faster. 

Make sure vents are done at least close to correct, and make sure bath fans and dryer discharge to outside, rather than into attic. 

or a judge will discharge it.

And then the water meter is also disconnected, so that there is no way for water to get into the pipes.Then, all the traps get anti-freeze - traps exist in lots of places (toliets, under sinks, under tubs, under shower pan, in the washing machine discharge pipes, in floor drains - I may have missed some).

HD sells a plastic discharge pipe that's threaded on one end for $3.