Hello Kevin,
Its definitely possible and its called a private app. It would communicate using the Shopify public API but using your stores private API key.
To prevent other Shopify stores from using it you just need a) to not submit to the app store, and b) keep the app link private/ under login.
You can get working with private apps right away, but we are not able to support the development or any coding.
There are not any specific restrictions, you can use any web technologies and you will be hosting it yourself. So for example while Shopify doesn't render any PHP on our servers, yours could.
Here is the basic page to get started: http://docs.shopify.com/api/tutorials/creating-a-private-app
About your last question regarding updating the stores DB, this is absolutely what our API was designed for so play around with it: http://docs.shopify.com/api
Have a creative weekend,
Alex Massaad