Watch whether an app uses TLS I'd like to inspect the network usage of my apps (e.g., a banking app) to determine the security protocols they employ (if HTTPS: TLS layer version, encryption protocol, key size, signature algorithm, if multiple cipher-suits are supported, then suit preference--mandatory/preferred FS support--etc.).

In a browser, that's easy; not so much for standalone apps, which for all I know might not be using modern protocols.

Are there apps that listen to- and analyze network activity for that information?
There are Android apps available that can act as VPN service on the Android device itself (even without root). Afterwards all traffic is redirected through that app.

The app captures and displays the proxy, allowing you to detect unencrypted data.

Available Apps:

Packet Capture 

On the step "SSL Decryption" select "skip", otherwise your SSL connections will be intercepted and decrypted.

Apps that use correctly protected SSL/TLS connections will fail while Packet Capture is running and capturing the data.