Are there any trace tools for android, like DrWatson for Windows?
And what about a firewall etc to check what internet connections are established?
Not sure if I get exactly what you’re after, but maybe OS Monitor is an app that addresses at least some of your needs?
There are also some development/debugging tools built in by default (System settings > Developer options). See http://www.androidcentral.com/all-about-your-phones-developer-options
Firewall: I use AFWall+ (on F-Droid) but there are others. AFWall+ allows you to specify per app which connections are allowed or not either on wifi, 3G, and roaming.
I’m not sure what you mean by trace tool - last time I used DrWatson it was a debugging tool. I think @Herve5 has it right, debugging is build in, though I not used it yet.
For network tracing, I have installed LanDroid and Fing. And instead of a firewall, I use the rights management system GravityBox (of the Xposed framework).