I have done a similar app like this, which is in fact an in-cab entertainment system. I have also written a blog post about it, you can check it out here: http://arnab.ch/blog/2012/01/android-auto-updating-homescreen-application/.
This is a complex application and let me list out the relevant items for you:
- Your app should be a HomeScreen application (search google for how to create HomeScreen app for Android)
- It seems clear that you would have some control over the device, so you can ensure that no additional applications are installed.
- The homescreen can be dynamically enabled/disabled, check out KytePhone app to see what I meant. In short you would need some password to exit your HomeScreen app.
- If you want to silently uninstall/install any application, then you'd need root access, or you'd have to have a custom Android build where your app will have System privilege (might not be what you're looking for).
I hope I am able to give you some direction, if anything is not clear then let me know.