Actually, your last issue is most likely with gatekeeper. Go to your Security and Privacy preference pane and select "Anywhere" then launch all the apps you commonly use. It resets itself after a period of time, so you might encounter it again with apps that you haven't used in the next 30 days.
Thanks for the advice, but I got the message when I try to open normal Mac applications like Activity Window (or how it's called in English) or Terminal or Google Chrome.