Is there a Android POS full node wallet apps that is open source? An app that can also stake? Are there any Android Proof of Stake apps that is open source. The peercoin app seems to be a light client. Some of the apps seem to be QT transformed to Android.
I think the short answer is no, there isn't an Android full node app. Being a full node would requiring downloading and storing the entire block chain, and that is too much data for mobile devices.

I don't know of any mobile wallets that stake. And you are right that many of the apps are just the QT code transformed to Java. I think it was done this way on purpose so as to ensure that the two references are bug-for-bug compatible.