First off, im not trying to tell you to do anything. im simply informing you of one method that works for me. to increase strength you need to increase the weights you are lifting. For instance, if you start walking more, you may burn fat, and you will gain endurance, but you legs will not get stronger because you are always walking with the same ammount of weight(your own body weight)
However, if you wear a weighted vest and walk, you legs will get stronger because of the weight you have added.
Sore and burning are two different things. burning is the acid build up which goes away after a relatively short period of rest. you get sore because when you excercise you build muscle, building muscle puts tears in your muscle tissue which is then healed by the protein you take in.
as for the last question... if you want to increase your vertical jump, squats will only help to a certain point. squats build strength. your goal seems to be power and acceleration. the difference being that strengh is how much, and power is how fast you can get there. for that i would suggest looking into plyometrics, or jump training.