Earlier, when I was using Firebug and opened new tab - Firebug was already activated there. Now, in Firefox 51.0.1, I need to press F12 every time to open the Firefox DevTools. How can I activate them by default for new tabs?
The allPagesActivation option in about:config doesn't help.
Firebug's activation model is URL based following the same origin policy, by default. It can be changed to be always on via the extensions.firebug.allPagesActivation property, as you already noted.
The DevTools' activation model, though, is tab based and there is no option yet to change that (as of Firefox 51.0.1). Though this is already requested in bug 1284012.