No. The open source drivers are better at almost everything. The only reasons to use the propriatary one is if you need some OpenCL improvements of if you are using a Radeon Pro GPU. For normal usage and gaming the open source driver will offer more performance and better compatibility.
As far as I can tell mesa and the proprietary drivers both use the ROCm packages for OpenCL. I don't think there's actually a difference on that front.
I honestly don't realy know. The Arch wiki says that there are some differences with AMF and OpenCL but I don't know how up to date that information is.
Mesa has its own OpenCL implementations for AMD GPUs too: Clover and RustiCL.
However, Clover is not really developed any more (afaik) and lacks some important extensions, such that many programs can't use it. RustiCL is rather new, and I don't know how well it works.
No, I think I have the library rocm smi lib (or something like that) (on Endeavour/Arch btw) installed and it is used by btop to display GPU stats.
EDIT: The Arch package is extra/rocm-smi-lib
I'm serious. Systematically install, test, and take notes on all the drivers (there are like 4 iirc). If I wasn't taking a crap right now, I could check my config and send it to you, but there is no guarantee it will work for you. Best thing to do is to do science at your own computer.