Emulators (FS-UAE & MAME) don't start at all and I suspect the problem is with a video card.

FS-UAE [amiga emulator] throws the following message before it gets stuck without launching its screen:
xrandr: Failed to get size of gamma for output default


Similarly Mame says this when I try to run it:
video_init: Initialization failed!

My video card is:
Advanced Micro Devices, Inc. [AMD/ATI] RS480 [Radeon Xpress 200 Series]


I have Debian Jessie with LXDE
My monitor is an early low resolution (1024x768) flatscreen

On the previous Debian version FS-UAE and mame started but especially mame was too slow to use (I think some NES emulator had the same problem). At least a Sega Genesis emulator (GensKmod) is currently working, so the problem is not all-pervasive.

Did you install the ATI driver?

^
No, I didn't.

Here's my Xlog which I think displays information on my video card driver:
http://pastebin.com/uEy8KKeL

rs480 - from 2004?

I don't even know if the OSS driver goes back that far. Even if it "Supports" the rs480, it may not have recently been tried out on one.

But it's Linux, so old hardware should work.

well, it does work, doesn't it?
it falls back to some default video driver, i guess.

but running specialised software...

i guess that those emulators need video acceleration, and you can't have that with those fallback drivers.
search the internet for your specific card and how it runs on a current linux system.
see if there's support for hardware acceleration, and if that model supports it at all.

edit:
please post output of

01:05.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] RS480 [Radeon Xpress 200 Series]
Subsystem: Hewlett-Packard Company Device 2a24
02:03.0 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL-8100/8101L/8139 PCI Fast Ethernet Adapter (rev 10)
Subsystem: Hewlett-Packard Company Device 2a24
Kernel driver in use: 8139too
02:04.0 FireWire (IEEE 1394): VIA Technologies, Inc. VT6306/7/8 [Fire II(M)] IEEE 1394 OHCI Controller (rev 80)

Work only started on the current AMD/ATI linux video driver when amd took over the reins at ATI. My 2007 rs600 card was declared obsolete (2009), and everything before it as well. The OSS driver applies, but they don't test old cards.

The other stuff in there gives age away as well. Some of those old boxes did not have interrupt sharing. Some firmware (e.g. the 8139 nic) did not do it, or did not implement it properly. It was necessary to find what interrupts a device was happy on and continue to use them. I lost network in linux around then because the pc was assigning the 8129 nic interrupt 12 and it wanted interrupt 11.

That sort of messing would throw emulators big time. Just to eliminate this, post the output of

i added code tags to your output.
please do so yourself from now on.

it is just as i thought - there's no dedicated graphics driver module assigned to your gpu (there should be a line reading "Kernel driver in use: ..." for your VGA device).

and businesskid's explanation kinda confirms my suspicions: a proper driver for this old card is not easily available; maybe not at all.

Your old hardware is indeed working.

Your game emulators are a different story, and you need to have more realistic expectations for them.