I have only a PCIe 2.0 x4 and a PCIe 3.0 x16 free and I need to install a 10Gbps SFP+ card and a graphic card for transcoding with Jellyfin (the CPU is an old E5-2620 v2).
Since I can't find any SFP+ card that is x4, do you know a graphic card with decent transcoding capabilities (I'd day 2 4K simultaneously) that is only PCIe 2.0 x4?
The absolute best bang for your buck new GPU's for decode/encode are Intel ARC GPU's. They use Intel's Quick Sync Video system which is some of the best supported encode/decode libraries out there, and they're cheap.
An ARC A380 is easily had for $110, runs entirely off 75w PCIe slot power requiring no additional PSU wires, and supports H264/H265/AV1 encode. It's a no brainer.
As long as it physically fits the slot, it should not have an issue with the lower PCIe bandwidth. The lower end GPU's really need very little even for video encode.
Resizable BAR was previously cited as a requirement for Intel ARC cards, but I think the drivers today can do without. Sounds like your system might be too old to have that. Might be a soft requirement, as in you'll see a performance drop if you don't have it.
Do you need to run the Intel cards with an Intel CPU to get the quick sync benefits? I upgraded my desktop last year and am going to convert my old Ryzen 5600x system to ... something. Not sure what yet though. Just working on my options.