Question #1:
I stumbled upon this link some time ago:
http://docs.oracle.com/cd/E17076_02/html/installation/build_android_jdbc.html
(Disclaimer: I haven't read the whole tutorial myself)
It tells you what's needed for getting the MySQL connector to work with Android.
Basically, from what I know, the standard MySQL connector for Java, doesn't work out of the box for Android. You'll need to build it on your own.
After building the connector, you can include it in your Android project, precisely the same way as you do in a normal Java project.
Your Question #2:
SOAP is a mere layer of abstraction, for not working directly on the database. If you use a layer like SOAP, you can swiftly change your database layer without affecting much of the communication from-and-to the Android app.