Your facts sound a bit unusual unless you are in a condo or home with HOA dues where there is some incentive for the bank not to foreclose for two years after your bankruptcy.
There are two separate questions you are really asking in regard to the timeline. One, what is the timeline. That is relatively straight forward and governed by RCW 61.24. You will receive a notice of pre foreclosure options letter. Next you will get a notice of default. 30 days after the notice of default you will get a notice of sale. 90 days after the notice of sale the foreclosure sale will be held. So typically you are looking at between a minimum of 120 days if you do not respond to the notice of pre foreclosure options letter to probably no more than 210 days if you do respond.
But the second and more important part of your question is how long can you stay? If the Bank does not send the notice of pre foreclosure options letter for sometime, continues the sale, or gets tied up in mediation can alter the timeline above.
My suggestion at this point is hope for the best but prepare for the worst. Watch for the notice of pre foreclosure options letter and then contact either an attorney or pro bono group for a consultation regarding your rights.
As for partial payments, there are actually risks given the length of time you are still in the house and how that interacts with the bankruptcy discharge. If you surrendered your property in your bankruptcy and have not made any payments and they have not foreclosed yet, why rock the boat and make them think there is money on the table? It probably won't help your situation and most likely will hurt it.
Paying part of your mortgage may not help much. Most lenders don't accept a partial. Talk with your bankruptcy lawyer.
If you are trying to stay you should contact the Bar Association at WSBA.org for some foreclosure help and perhaps applying for some of the programs will stall the ineveitable. The automatic stay from bankruptcy will also stall the process until discharge or dismissal.
The timelines are fairly short for a trustee's sale and after the notice of default (30 days) appointment of a trustee and notice of a sale (90 days) then a sale can be held. If you demand all of your rights such as the face to face meeting with the lender, the mediation and applying for all programs and modification avalible you may extend the time.
Only If and until you and I sign an Agreement for Legal Services, I am not your attorney. These answers are provided for informational and/or novelty purposes