1. I would go Cat6 for sure, especially at that distance (175' is about half the max distance to run any PoE)
2. 100m typical limit for full speed/power
3 & 4. That sounds like it would work, personally I would go all unifi and use their stuff - they have some outdoor routers that would probably work better than going to the shed (unless you want the wifi in the shed)
https://design.ui.com/wizard You can design your place there and figure out what you would need for coverage.