To the OP I'd say you are pretty sound in your thinking all the way around.
The 7mm Rem is an awesome coyote caliber if you are not trying to shoot them for pelts and even if you are, you can get non expanding solids that won't blow them up too bad as long as you avoid head and shoulder shots.
The 700 Rem action is the most popular action ever at least in N. America so again, sound thinking.
I will say though that the .260 Rem would be even better for your applications economically as would the good old 6mm Rem.
Barrel live with the .260 and 6mm are virtually unlimited unless you are running enough rounds through them in quick succession to really heat one up. With the 7mm Rem, barrel life is generally good but not as good as these two.
Another thing to consider would be to see if you can find a Model 70 Coyote in 7wsm, a fantastic alternative to the 7mm Rem.
As someone else said though, don't dismiss the 6.5's/260/.264wm. For your application the bullet selection and ballistics are fantastic.
Personally my favorite coyote gun is a 7mm STW but then I like watching them pop like prairie dogs hit with a .220 Swift LOL.