Not only wouldn't I pay to have strings changed, I wouldn't even let someone in a music shop change them for me. I once saw a shop proprietor tapping bridge pins in with a hammer! The pin doesn't hold the string it simply prevents the ball end from slipping out of a notch in the bridge plate below - although with constant changing the notch can wear out.

Its difficult to estimate how much slack you need in order to get the correct number of windings around the post. One thing I learned by watching a luthier fitting strings was - (i) roughly line up the hole in the tuner with the slot in the nut (ii) insert string into hole and pull to tension (iii) keep some tension on whilst pulling the string back out of the hole a measured amount using the frets (I usually find 1 to 1.5 frets enough - more for lower strings) and bend the string through 90 degrees (iv) wind up to tension.

It matters not whether you change them one at a time or all at once. I find the latter simpler as you have a reference from which to tune.

If you don't like changing strings, don't buy a mandolin!