All mortgage companies want all past due debts PIF. Nothing lingering. DTI is the main engine for approval or not.
You have two options:
1. You could see if T-mobile can/will accept the payment, and have them recall the debt from the collection agency. They may be able/willing to do this assuming they still own the debt and their arrangement with the collection doesn't prohibit it. In this scenario, the collection agency would no longer have reason/authority to collect the debt and would remove their tradeline.
2. You could try to arrange a pay for delete with Convergent. They may or may not be willing to do this; I've seen mixed reviews. If they do, it will be deleted after payment is made, generally within 60 days. If they don't, then you'll probably still have to pay it in order to obtain a mortgage. Most lenders won't lend with potentially liabilities like this out there. But you probably won't see any/much score improvement until it falls off your reports.
To answer a few of your other questions.
Credit monitoring services showing it at 10 months old is plausible - and likely not illegal. As long as the collection agency has collection authority, they can add or remove the tradeline from your reports whenever they want. I'm assuming what your referring to is when the collection was added to your report.
Most traditional mortgage lenders will require the debt to be taken care of. But you should ask your lender.
If T-Mobile agrees to recall the collection, they likely won't do so until you make the payment. They may or may not be willing to put that arrangement in writing. If not, you'll have to trust them. You could record the conversation, but stay within the laws of your jurisdiction when it comes to recording people. Nothing is implied, be very specific with what you're asking/offering.
