January 7th 2014 Edit
I currently have a PCIE x4 Intel I350 NIC in a PCIE x16 slot meant for graphics on a Intel DH77DF motherboard working without a problem.
The original post below stating that "as long as it fits it should work" still stands. It's still possible that some motherboard manufacturers have a PCIE x16 slot that won't accept anything other than a graphics card. I think this only happened in the early days of PCI Express, and modern motherboards (say Core 2 and newer, and definitely Core i-series and newer) are pretty flexible in what you can use the PCIE x16 slot for.
Also, make sure you have on-board or on-chip graphics, or a video card somewhere. Most motherboards won't boot without video. Although, sometimes you might be able to configure the BIOS to ignore the lack of video and continue booting.
I figured I'd return to clarify my answer after encountering this very situation myself :)
Original Post
Initially I'd say there's no real difference between the PCI-Express slots and as long as it fits there's no problem.
However, the problem is if the PCI-Express slot is specifically labeled for video card. The tomshardware link shows the user placing the TV Tuner card in his second slot, not the first. The first slot (in your case, your only slot) might be graphics only.
Some point to yes. Some point to no. It's hard to tell and seems like it's up to how the motherboard manufacturers decided to integrate the chipsets.
Well, graphics cards are routinely run on 8x speed when SLI is in use - there will likely be a small performance penalty, but I wouldn't think its huge - else SLI wouldn't be worth it.
Its a bit apples and oranges but with a different (and I believe newer) graphics card - The AMD Radion HD 7970, Anandtech showed very little practical difference between x4, x8 and x16 in PCIe 3.0 (Yes, its a newer version of PCIe as well. Finding these benchmarks is hard)- since PCI-e 2.0 is half the speed look at the x4 graph for x8, and x8 for x16 . While yours is nvidia, Its a lot older and would need less bandwidth.
I'd also suggest doing a benchmark with some tool that compares it against other similar systems and getting a definitive answer. I'm guessing it should work acceptably under most workloads. Considering the scenario, it does seem worth doing.
Best Answer
Yes, technically, it should work, as long as you don't have another card in the PCIe x16 slot on that motherboard. Though the PCIe x16 is usually used for graphics cards, the standard indicates that any PCIe card is supposed to be able to work in any PCIe slot which is that size or bigger. The expansion card you show is a PCIe x1, which is supposed to work in a x1, x4, x8 or x16 slot. But, as with all standards, support by motherboard and card may vary.
You can read more about it http://en.wikipedia.org/wiki/PCI_Express.