I've read a few of the questions and answers on working with someone when you are repaying a credit card. I read where someone mentioned to get it in writing if they accept payment that they will report the account paid in full and remove all derogatory items. I was curious, do all agencies have the ability to make this agreement? I remember one time being told that they could not, do you think they were only saying that because they didn't want to agree to that part?

Most data furnishers agreeements with the credit reporting agencies stipulate that they will not delete accurate information. If you're dealing with a collection agency, you can ask for a pay for delete. If they won't agree to that, then you can try to go with a Non-Disclosure agreement(NDA) where you pay them and then dispute the tradeline on your credit report. They agreed by signing the NDA that they would not verify the dispute. No verification=automatice deletion on your reports. You have to make sure that you add a stiff penalty like $10,000 for reverealing the terms of the agreement.

Can they, yes of course they can, will they, no many will not.
In fact many will not even agree to a good settlement. Each company has their own rules, and also each collector has their own agenda and quota that they are working on

Collectors fall into two categories; agents and creditors. The creditor is the actual owner of the debt. An Agent is someone hired by the creditor to collect on the debt. Frequently, the collector will hide whether they are an agent or creditor or even lie about it.

But an agent may not have the authority to agree to a settlement or their authority may be limited. And its up to the creditor to report the status of a debt to the credit bureaus.