With that out of the way, you have already completed the negotiation phase of the process. They can't reopen the salary question without rescinding the offer completely. Why would they do that? They've already invested heavily in recruiting you and have offered a salary they must be comfortable with. To squeeze you now would make no sense. If they did that, you wouldn't want to work there anyway.
The third party is certainly some 'bcheck' company that services many clients that do require salary info as part of their negotiation process. They don't have a custom process for BigCo, so you got the generic form everyone gets, regardless of employer. They run a series of standard online checks, maybe call past employers and get only dates and title (no employer gives out salary data when asked), and forward the whole mess to BigCo HR. The bcheck company has no opinion in the matter. I'm guessing the only use for past salary data they might have is to judge how they are doing versus the market.