When I launch an app, the focus doesn't move to the opened app
You can configure it with CompizConfig-Settings-Manager.
1) Install it: sudo apt-get install compizconfig-settings-manager,
2) Open ccsm (Alt + F2 and type ccsm and hit Enter)
3) Go to "General Options" > "Focus & Raise Behaviour" and set "Focus Prevention Level" to "Off".
4) Enjoy ;-)
If you are feeling lazy, just paste this at terminal and you're done:
gconftool-2 --type=Integer --set /apps/compiz-1/general/screen0/options/focus_prevention_level 0
No it is not correct behavior. The answer to the question "why is it happening" is because there is a bug. I suggest submitting a bug report. Surprisingly I cannot find an existing bug report for this, though I can confirm that it does happen. Post a link to the bug when you have done it so I can attach myself to it.
Edit: Found it. Bug #781931 Not sure why this answer has been down voted as it is certainly the answer to the question of why this happens. The work around using ccsm is not an answer to the question, it is a workaround for the bug. That in itself is very useful and worth posting but it does not answer the question that was asked.
The command above didn't work for me and I didn't want to install CCSM, so I did some digging and found that the following command did the trick (I'm using Ubuntu 15.04):
dconf write /org/compiz/profiles/unity/plugins/core/focus-prevention-level 0
Edit: This solution still works for 16.04