Why can't I just use the word 'walk' instead?
Actually, you can use "walk" instead, and for everyday usage, you should. This definition of "pad" is technically valid, but it is specialized and rarely used.
How does the word pad differ from walk? What difference does it make?
When I do see "pad" used this way, it is almost always referring to an animal, not a human. And not just any animal, but one with soft contact surfaces (i.e. pads) on the feet, such as dogs or bears. You would never use "pad" to describe the movement of a horse or chicken.
When used for a human, "pad" refers to a quiet and/or stealthy walk, not just any walk.
Does the word fit in my sample sentences?
Not quite. If you do find a case where it makes sense to use pad, imagine how the sentence would be written with "walk" and then use the same form of "pad" in instead.
* People are walking the street
is incorrect because it is missing a preposition. Instead, you might say
People are walking/padding down the street