Butter is fantastic for scrambled eggs, fried eggs and some vegetables that do not require high heat.
Good-quality ghee is the go-to fat for high-heat frying though, as its smoke point is much higher.
I fry low-temp stuff in butter (eggs, asparagus, mushroom, etc...) but frying a steak or a stir-fry, you usually want to crank up the heat. This will basically burn the butter. Still tastes good but you've oxidized the PUFA's (killed your omega 3) and denatured lingering milk proteins. Ghee, tallow, and coconut oil are good for high-heat frying. Olive oil is somewhere in the middle.
Olive oil + butter is a classic cooking oil combination.
In theory the smoke points are a bit low unless you're frying something like eggs. Ghee in theory would be a better alternative.
Both. The coconut oil keeps the butter from burning, so you get the benefit of taste and pan lubrication without burn.
Butter is absolutely fantastic! As long as it's a grass fed organic butter such as kerrygold. (http://kerrygoldusa.com/products/butter/). I tend to prefer the taste of frying something in butter over coconut oil simply because I do not care for the taste of coconut, which does come through if you are using unrefined coconut oil. However, if you enjoy the coconut taste I think either the coconut oil or the butter would be great. They are both healthy for you. You could try the different options with different meats and veggies to see if you prefer different flavor combinations.
Good luck and healthy eating!