RE: But I'm not sure the payroll reporting that reports on these totals will be messed up if I change them. ????

You can change the liability account on payroll items to a different liability account. It won't change the way they appear on payroll reports.

RE: I also realize that I will need to update every employee in the Payroll tab for these new Item names. 

There shouldn't be any new item names to update after changing accounts on payroll items, which doesn't create any new payroll items. Editing existing payroll items generally does not require you to edit the employees.

RE: However, will I have a problem at doing the quarterly or annual tax reporting? Is this better left to do only at the beginning of a year?

I don't think it matters when you change the accounts on the payroll items, especially if you update all of the transactions retroactively with the change (which is an option you'll be offered after changing the account when you save the item.)

RE: I don't know how QB selects some of the GL #'s for the reporting for taxes, or if those numbers aren't meant to be changed. Is there a screen that describes more detail in that?

Taxes are reported by payroll item, not GL account. QB has a default account, usually named "Payroll Liabilities" that is used on taxes and some other deductions automatically when the items are created. But you can change it any time you want and it shouldn't cause issues.