Welcome to the forum!
As for your questions:
1. When we call insurance carriers, we identify ourselves as 'calling from Dr. Soandso's office'. It is too complicated and confusing to say that we are a billing service and blah, blah, blah. If we are asked directly if we are in the office, or if we are a billing service we always answer honestly. We are not trying to hide anything, it's just too complicated to explain for every single phone call.
2. We have a few providers that we access their software to obtain the info we need to bill, but we always use our software to bill. It would be too difficult to learn a new software for each dr. We are used to ours, we know ours, we have our electronic systems in place, etc. So it wouldn't really be feasible for us. However, if you are just starting out, and that is an option that will save you from having to purchase software immediately, it may work for you.
3. When you are doing strictly 'clean up' work it is harder to make money. Your regular fees aren't really enough. For example, he's giving you the problem claims so there is a possibility (higher than normal) that the claims might not all get paid. Your risk is higher than if you were billing everything. Also, like you said, it will be difficult to track. You might want to consider charging a different way on those claims. Either a higher fee, or a per claim, or hourly. In any case, I'm not sure what to recommend for tracking other than a manual system, which is not ideal.
Hope that helps you out. Good luck in getting started.
Michele