No, running AGPL programs does not require you to share the code. Modifying AGPL programs requires you to offer all users a copy of the source code. You can for example modify the code and offer it in a public github project and after you have made the modification make the github project private.
Don't worry, this is a common misconception. Just running AGPL code doesn't obligate you to do anything
In general, no you don't. You only do if you let other users interact with the AGPL'd program over the network.