The linked drive may use a PCIe interface, but it not a PCIe form factor card. It is an M.2 form factor device and requires an M.2 slot. Without some sort of adapter, it will not work in your M.2-lacking motherboard.
The M.2 form factor is designed to connect to multiple interfaces - SATA and PCIe - over the same physical connector. Depending on the capabilities of the connector and device, they're keyed differently to prevent you from attempting to connect incompatible devices (a PCIe-only device in a SATA-only M.2 port, for instance).
The adapter you linked should do the job. As far as PCIe 2.0 vs. PCIe 3.0 goes, PCIe 3.0 devices are backwards compatible, they'll just run slower.
As far as lane count goes, you can use a card that demands fewer lanes (4x vs 16x) in a slot with more lanes; the other pins aren't on the card, and therefore do nothing. Going the other direction requires paying more attention. Some cards need the bandwidth of all the lanes they're spec'd for, and some boards have things like 8x electrical slots in 16x mechanical slots for security/fit purposes (there are missing pins on the board end).