Will vs Would? Can both of them be used for future [closed]

Both your sentences are grammatically correct and both have a future meaning: we understand that she hasn't done anything yet and that she is unlikely to do anything in the future.

Your other sentence,

The lady listened carefully to my complaint but I doubt she will do anything about it.

is very possible. I guess your concern is about using past and present in the same sentence. It is alright to do that, since the two actions do not both take place in the past. Think of:

I know (now) that the lady listened carefully (in the past) to my complaint, but I doubt (now) she will do anything about it (in the future).

"Will" is more of a definite future mood, while "would" is a conditional mood, which usually depends upon some other "if" condition.

For example:

"...I doubt that she would do anything about it (if she had the chance)".

On the other hand...

"... I doubt that she will do anything about it"

suggests that she is perfectly able to do something, but she just won't.