I'm a little rusty on my residential lending, so bare with me.
If you are applying for an FHA loan, most lenders will accept another lender's appraisal. If you are dealing with a conventional loan product, or a portfolio lender, you may run into issues since most of those lenders will want to order their own appraisal.
In the case where the lender, not the broker, and I don't think brokers order appraisals much anymore, ordered the appraisal, it is at the underwriter's discretion if they want to release the appraisal, and to honest, an UW doesn't care about saving you any money.
Again, I'm a little out of touch when it comes to residential lending, but I did a little checking into this issue, and from my digging around, this is the condensed version of the info I read.
As far as your broker not telling you who they're dealing with, that's an issue between you two. If you have a good working relationship with them it really shouldn't be an issue to disclose to you who the lender is.
Many borrowers don't care who the lender is as long as they're getting what they want, but in your case, perhaps there's more to a loan than just the rate.