Electronic – Why diameter of telephone cable is exactly 0.44mm and 0.63mm

In theory, this should work. Cat6 cable is generally fairly high quality, low capacitance, low resistance, and high bandwidth. The twisted pair and individual shielding also helps. Host powered USB Extenders can use cat5 for up to 100m.

As for the capacitance for the i2c bus, you can get that information from the datasheet for the cable. Typical Cat6A capacitance is 330 pF per 100m. Look at this datasheet for an example.

To be honest, 6 meters is short. You shouldnt have any issue.

You could probably go the same route that Power over Ethernet (PoE) goes: Have the supply voltage be a DC offset on pairs of the cable, and couple in your receiver through transformers with a center tap on the biased side.

5m sounds a bit much for LV differential, but that really depends on your equalizing abilities.

Also, at some point, the overhead you're incurring by extensive cabling and dedicated differential transceivers might be more than it's worth – assuming your application could withstand the latency, have you looked into using one microcontroller on each end of your 5m distance, translating the unidirectional signals, clock and chip select to a bidirectional bus? I'm thinking about a really minimal system based on USB (that's the cheapest long-distance bus that comes to mind). Or, CAN, which is actually designed for this kind of problems. For both, there's cost-efficient MCUs that come with integrated PHYs.