If you buy Mathematica, do you still have to pay $5 per month - as you do for wolfram-alpha - in order to access more than three step-by-step solutions per day?
Do you mean, using the WolframAlpha connection from Mathematica?
Mathematica does not have step-by-step solutions built in.
I just tried, and was able to perform more than three (five or something) step-by-step queries in a row. This option is available through == WolframAlpha connection. I have no idea about WRI policy on this.
There is a limit to the number of Wolfram|Alpha queries you can make per day in Mathematica. However, that limit is very high and rarely ever exceeded unless you were to write a program to try it.
So you should be able to ask for step by step solutions essentially as many times as you like by using Wolfram|Alpha through Mathematica.