These are really good questions, let me answer them separately but start off by noting this: You do not need to use starling in haxe to for GPU accelerated performance. I would only recommend using starling in haxe if you are porting an existing AS3 Starling project or for the sake of familiarity.
does the Haxe/OpenFL/Starling GPU accelerated performance match or exceed the AS3/AIR/Starling GPU accelerated performance? I use a lot of images permanently moving on the screen, and GPU acceleration is a must for me. I'm already pushing AS3/AIR/Starling to its limits, on high-performance Apple devices.
Yes, yes yes!!!! Performance is the big winner here. Our apps are as much as 10x faster and we have been able to render more objects at much higher frame rates! Not only that, but keyboard, mouse, touch, and gamepad input is much more responsive.
Does Haxe support socket clients/servers, and is there any cryptography support? (RSA, AES, Hash/HMAC, certificate parsing and verification)
Absolutely!! openfl.net.socket supports client side sockets (and websockets on html5) and familiar libraries like https://github.com/soywiz/haxe-crypto have been ported. Hopefully we can expect some better crypto libraries in the future, but these are way faster because of haxe.
For server sockets and UDP you can use the sys.net.* classes.
does Haxe have any kind of scripting support, to run code on device, at runtime, received over the web as text?
You might feel like giving this a peek:
We use a custom in-house script engine dubbed DimensionScript for server related applications for now. It has not reached full maturity but we intend to release it publically at some point.
I am glad to see so much interest in haxe. Please feel free to pick our brains.