In most cases, you’ll use an expense account for purchase transactions (enter bills, write checks and enter credit card charges).  However, if you ever need grant tracking you should use items for grant-related purchases as well.  In this case, you’d set up additional items for the grant phases you want to track.  Then you can use estimates for multi-year grants and the Job Estimates vs. Actual reports for grant tracking.  You should use classes on all your purchase transactions.

Classes don’t work very well for balance sheet transactions, so there’s really no reason to use a class on transactions that are booked to balance sheet accounts.  However, there’s no harm in doing so and it gets you into the habit of always entering classes.  There’s also a preference you can set to remind you to enter a class if you forget (Edit > Preferences > Accounting > Company Preferences).

And yes, expenses for domain names and tax exempt status would be G&A.  You’d only use one of your other classes when the expense qualifies for a that purpose.  For instance, a grant-related expense would be booked to the associated temporarily restricted grant class.