How to automatically repeatedly dial a number and automatically press numbers each time?
There's an app called Auto Redial that you can configure to keep calling the same number over and over.

I configured it like this:


(click screenshot to enlarge)

Note that I added ,1,1 after the number. This makes the phone automatically dial 1 twice, once the call is connected. This works even outside of the app, while dialing a number manually or from the contact list. 

I've set the hangup time to maximum because this particular service hangs up automatically when you don't get through. In that case, I can listen to the phone once in a while to check whether I got through yet. 

You may need to tweak these settings a bit depending on the service you're calling. 

Disclaimer: I have no connection to this app whatsoever. The problem described in the question was a real problem I was facing, and when I found the solution I decided to share it with Android Enthusiasts. I have no other motive with this post, and I'm not involved in app development in any way.