I have a box full of firewire cards, ranging from ISA to PCI and PCIe and a mixture of both 400s and 800s. It is impossible to say what card works with which system as there are so many variables from different motherboards to CPU/GPU and the firewire cards themselves.
I hope noisefleur will agree with me on this, its pot luck, good luck, but generally all combinations should work in theory, although not always in practice. There is no need to waste money on expensive specific cards, I have found using various cards, usually any and all combinations work. Whether it be TI VIA or NEC chipsets, it does not matter.
As for a successful install lets consider the steps in sequence.
Build computer/update all drivers
Install firewire card boot system and make sure it is installed correctly. SHUTDOWN
Switch on all firewire devices and then boot up system.
Install most up to date drivers available if any, specific to the system build x86 x64 Windows 7/8/10 etc
For audio interfaces make sure you have set the sample rates correctly and the master clock sync is set to the right local device.
This is a mistake I made when I first tried to connect my Behringer ADA8200 extension cos as I said before its only 48khz unlike my Focusrite Liquid 56 which goes to 192 so I had to drop the sample rate to 48kHz to make them work when switched on and connected together. I also have to make the Liquid slave and AD8200 master clock syn. Else I just get flashing lights and they won't work. Who reads manuals? Learn the hard way trial and error !!
Before they invented PnP (Plug in a play) motherboard and devices, we had to manually set DMA and IRQ system resources, now hopefully all modern systems do that for us and all works well. One issue that can be a right pain is if you have installed a sound card and not using the internal audio or HDMI. I have come across this issue with the creative labs SB0460 which I used to use in all my systems that were binned after I moved to HDMI. Now I just disable the internal audio I do not use it so do not need it and it cannot cause any issues, I just use HDMI (AV Amp) or USB to my audio interfaces.