Any maternity related costs are usually covered by your Krankenkasse, your health insurance. Typically one requirement for holding a long-term visa is the enrolment in a healthcare plan; you either had to proof that you have a foreign health insurance which covers your costs while in Germany, but since that's rather unlikely over the long term, you most likely had to sign up with a German insurance provider. Perhaps your wife's employer did this, or you've signed up for a freiwillige gesetzliche Krankenversicherung or a private insurance yourself. Note that one of you may also be covered by the other's insurance (familienversichert); though that usually only covers unemployed family members.
Pretty much any German insurance should cover any maternity related costs, including visits to the doctor pre and post, hospital stays, a midwife and other related things. There may be a certain Selbstbeteiligung, an amount you have to cover yourself before the insurance kicks in and covers the rest; that's usually the case with private insurances. What a foreign insurance may or may not cover is up to it; either way, check the details of your healthcare plan.
If you're paying taxes within Germany (which you should be, if it's your primary place of residence), you should also be eligible for Kindergeld (monthly payments for each child), Elterngeld and perhaps other -gelder. Check with your local municipality.
The German maternity leave rules pretty much only apply to companies operating in Germany. If your wife's company is entirely operating in the US and you just happen to live in Germany, German rules don't apply, US rules do. Which likely means you get little to no maternity leave. If your wife is employed in some form under German rules (perhaps a local branch office), German rules may apply. First and foremost check with the company, secondarily inquire with your local municipality for more information, or information about whom to contact.
(I am not a lawyer, but my personal situation is somewhat similar, and these are my experiences and gatherings.)