I'm a US citizen with a home in the States. Is it possible for me to get a "green card" in Germany so I can stay for an extended time, maybe even buy a home there as well?
Also, it isn't possible for me to obtain a German citizenship and still keep my US citizenship?
Different countries have different notions of residence, sometimes even several of them for a single country. For example, one country might decide you are a resident for tax purposes because you own a home and draw most of your income from that country but do not qualify for naturalisation or some benefits like free tuition for your kids because you are not physically present in the country most of the time. That means that depending on the definitions, it could be possible to be a resident in two countries for some or most purposes.
At the same time, residence permits for foreign nationals very often have physical presence requirements that make it difficult to retain them while spending significant time in another country. Case in point, you can lose the permanent right to reside in Germany (Niederlassungserlaubnis) if you are out of the country for six months. So keeping a home in the US is okay, going there for a couple of month two or three times a year too, more than that and you would have to restart the permanent residence process from scratch.
Note that I glossed over some details and just took the Niederlassungserlaubnis as an example, I don't know that you would qualify for that in your situation, at least not immediately.
