Carry a set of disposable gloves with your spare tube. Won't help you get the tire on any better but at least takes the dirty and maybe some of the cold wet part out of the equation.

Can't help with a tire suggestion as I'm a gaterskin guy, am very happy with puncture resistance, and get them on easily using my hands. (700X28 wire bead - I have noticed some rims are tighter using the same tire, so maybe that is why your experience is different)

If you flat frequently and can't find the cause change your rim tape to the Velox tape if that's not what you are using.