Firefox own developer tools won't log ajax calls in console. Firebug does. This and the fact that you need to open the settings tab to enable preserve log are my problems with Firefox tools keeping me on firebug.
Personally I've switched back to dev tools in the developer edition, mostly because the ability to see which elements have events attached from the markup view is very useful.
yes convenience, saves time switching back and forth. chrome didn't have this feature either but they recently added it (even though not as good as firebug's since when you click on the entry in the console it takes you to the network panel and hightlights the entry rather than opening it up automatically)
Hit the little arrow next to it. You can select Errors, Warnings, and Log. Log makes xhr network requests show up in the console (just like the network tab). I don't know what you mean by "fills up the console".