Cat6 cable is fine. I have taken a wire out of a window, round 2 sides of a house and down a floor, with the wire just attached to the building. Has been running ok for 5 years. The house in question is a very large 3 story Victorian property.
I would consider putting a second router, set up as a switch, on the end of the cable but would not use a repeater.
Your speed won't make a difference to the signal at the cottage.
I am not sure about burying the cable, it would need an outer casing for protection.