How long does the prosecution have to respond to a motion for discovery?

It depends on what was included in your discovery motion.  Some things, like witness statements, are to be turned over within 14 days of the request being filed with the court.  If you ask for items, like criminal histories that are not included in Alabama's discovery statute, you may have to have a hearing and compel the State to produce the items.  The 14 day timeline is a guideline-- it can be expanded or shortened by the judge depending on the situation.  Even after the set timeline, the State has a duty to supplement their discovery responses -- which means that you could be finding out new details the Friday before a jury trial.  If there is something that you need, you should request a hearing on your motion and get timelines set by the judge.  If the State files a late response, you need to ask for a continuance to prepare in light of the new information (if in fact you do need the more time to prep).  If you fail to ask for the continuance, then you waive the right to complain on appeal.  At trial, if the state tries to introduce evidence that was not disclosed as required by a discovery order, then the remedy is for the court to exclude the evidence-- but the decision to keep or exclude rests with the judge-- it's not an automatic exclusion.  So, as to the second part of your question, your charges will not be dropped because they fail to comply with discovery requests.