Are the following two sentences correct? Does the comma need to be there?
Walking down the stairs, she sighed.
When he told her she had missed a form, she sighed.
In this case, you use a comma to prevent the sentence from being misread. The sentence reads easier with the comma there than without it, so I believe you're two sentences are correct.
However, there is an easier way to write them that would increase their clarity. Instead of putting a phrase before the subject and the verb, put the subject and verb first, followed by the phrase. It's just easier to read.
She sighed as she walked down the stairs.
She sighed when he told her she had missed a form.
You obviously do not have to write this way; it is just easier to read in my opinion.
Yes, use a comma after an introductory phrase.
Walking down the stairs, she sighed.
But
Walking down the stairs was her main form of exercise.
The second example is trickier, because the introductory phrase is more complex.
When he told her she had missed a form, she sighed.
But the main part of the sentence is still "she sighed"—the introduction just tells us how it happened.