The staining on the back of the print can happen if you have undissolved amidol in the tray. Some people filter if this is a problem, and others just develop prints face up, and it isn't a problem, as long as it stays on the back of the print.
I'm sure my developer has reached those temperatures, and it hasn't been a problem.
I think that in addition to contrast control through waterbath processing, you get deeper blacks with amidol and Azo, and that makes the whites look brighter by comparison. I'd used other developers with Azo before, but once I tried amidol, I haven't looked back. I tried Weston's Amidol in the kit from Photoformulary first, and that was a significant improvement over Neutol WA, and then I ordered bulk chemicals to start mixing Michael A. Smith's formula for Azo, and that was even better.
Smith's amidol formula for enlarging papers works well with some older-style enlarging papers like Efke Emaks, and I suspect it would do well with Slavich Unibrom.
If you want to make contact prints and enlargements in the same darkroom session and stretch your amidol, what I do is do my Azo prints first, then add the requisite amount of KBr and benzotriazole to the tray to convert it to the enlarging paper formula and finish the session with enlargements.
Smith's formulas are at
www.michaelandpaula.com.