So is there a way to validate our css .So that it will work properly in all the browsers during editing process in content center ?
Not to be rude, but the only way to validate CSS beyond syntax is trial and error (open a page in edit mode and see if you messed it up). I don't know what other response you were expecting.I would suggest however that you become familiar with the differences in structure of the edit mode and understand why your rules didn't work in firefox, so that you can avoid the same issues in the future. Edit mode injects a bunch of additional wrapper divs and objects to make those dashed outlines and "object" handles happen, and there's many more CSS includes for the SP UI that cascade into your content pane and mess things up.