Does it work out of the box?  With some tweaking?  At all?

No out of the box, since installation is needed. No particular tweaking needed, should work after proper installation. Here is picture from Devuan Jessie:

This is from low power Kabini APU, your's will differ of course. Official upstream FGLRX supports kernel versions up to 4.2 and xserver versions up to 1.17. So default Jessie 3.16/1.16 combo is expected to work, no problem. Kernel version support can be patched further, but xserver can't!

So if you patched it to 4.9 kernel it should work there, like you see in picture i am running latest vanila longterm 4.9.33 kernel there... Same thing should work on Devuan Ascii with default 4.9 kernel, but only with downgraded xserver version since there is 1.19 default which is not supported, etc...

Basically only attention should be what major xserver/kernels are running and what chips particular FGLRX version supports.

15.9 found in jessie repo supports EG/NI/GCN hardware also, but 15.12 from backports only GCNs... on AMD site you can found Catalyst-PRO driver which supports workstation chips that are listed there, etc...

Of course people with GCN 1.3 or Gen 4 AMD hardware, so called Polaris or later shouldn't try this at all as that hardware isn't supported by FGLRX for sure.
So up to GFX7 totally, but GFX8 partially - some GFX8 are supported and some not at all, etc...

OK, i think that info is enough So does it work? Well it does, but it depends what your chip is, what are kernel and xserver versions. Deal is the same on average for any distro.

Lxrandr doesnt save the resolution or the overscan settings across reboots for many years now.

Yes it does save the configuration you set. And then it hides it from you unless you're running lxde. Edit ~/.config/autostart/lxrandr-autostart.desktop and comment out the line that hides it from you.

#OnlyShowIn=LXDE

If you change it and save again, you'll need to edit the file again.