So I think the power is the least of the problems. First of all: A x4 card in a x1 connection does not work, although there is otherwise a lot of (downward and upward) compatibility in the PCIe area. The connections are simply not suitable, namely the wrong way round. In addition, this makes little sense for a TB card, x1 usually allows 10Gbit, in a brand new PC perhaps 20, which is too little for TB. Everything that is interesting there - external graphics, SSD - will run much too slowly or not at all.
Second problem: In general, these things only work if this particular connection is available on the mainboard (usually "TB-Link" or similar). That's probably what you mean by the 10/12 pin problem in the question; I wouldn't have much hope, they'd rather sell their own solution. If you get the PINOUTs, you can reconfigure these pin connectors, but it's fiddly and nobody knows whether it will work or whether something will break.
A TB card like this is actually also available from Lenovo for their own workstations. It tends to be a bit more expensive than this one (I mean from 130), but it also works. Maybe you can get a bargain, Lenovo sometimes sells the accessories cheaper.

No, it is treated like a graphics card and requires the same amount of power