Writing my first app for Andriod and I have a few questions

Thanks in advance for taking the time to read this.

I am a developer with 17 years of code experience. I have some code that I've written over the past few years in my spare time and it's heavily dependent on a postgres database. I want to port this program into a smart phone app and have a ton of questions but the most glaring questions are:

1. I can organize the app to use a local database on the smart phone but I am wondering if it would be better to keep the large database data on a server with only the users access credentials stored in the phone app.
2. I am just an independent developer, Is it possible to set up a server to host this database at my house serve up it's contents in response to requests from the apps on phones? Are there any web hosting solutions that are affordable that also allow a developer to customize the database easily?
3. Is the sqlite database that comes with an Android robust enough to handle multiple tables some of which holding thousands of rows of data?

I could invest in a small server and set the database up on it.. and even configure it to use my existing network connections but I am not sure this is the best approach.

ANY suggestions or advice you can offer on this topic will be appreciated a great deal. I really want to do this right from the start.