It depends on the issue.

Chrome DevTools are a toolkit which helps troubleshoot issues, can can clear up issues in other clients, assuming they are having the same issues.  Firefox and IE have their own dev tools (but they don't offer spiffy interactive video courses for them).

Currently, Chrome uses Webkit (soon to be Blink), Firefox uses Gecko, IE uses Trident, and Safari uses Webkit.   Each of these different rendering engines may process CSS slightly different.

In regards to page loading, a lot of what Chrome identifies will fix the issue across clients.  That said some bugs are client specfic, such as older editions of IE incorrectly rendering the margins of elements (and this is why developers often charge more for supporting IE 6 and IE 7).  Those would in some cases have to be fixed on a client by client basis.  You can get around a lot of that however by using standards compliant markup.

It's pretty clear from the Waterfall timeline however, that the server is taking longer then it likely should to process / return requests in regards to loading webpages and their dependancies.