Login to AppStore over cellular data, jump on public wifi, download/install app is "safe"? Is it "safe" to install an App as such:
(1) using only a cellular data connection (wifi turned off) login to the App Store.
(2) enable wifi.
(3) connect to public wifi network.
(4) disable cellular data.
(5) download app.
(6) enable cellular data.
(7) disable wifi.
(8) install app.  

The iPhone can jump between wifi and cellular data without breaking a tcp connection right?
Your scenario won't work I'm afraid, because a TCP connection is created between a source IP address and port, and a destination address and port.

In your scenario, your phone's source IP address would change when switching network interfaces. This would break the connection, so the App store will not recognize that you already connected before, even though you securely connected over TLS (HTTPS)