If these balances are left over in these types of accounts, first, you should try to find out why these amounts are left over, and then, move the money appropriately, however, if you are unsure what to do with these balances, then moving them to an income or expense account after you've ran your year end reports, but before you've ran the year end utility will make sure that the balance is zeroed out in those accounts at the start of your new year. By doing this, you can then inactivate the asset/liability/dedicated account in question.
Yes, you can enter transactions for your new fiscal year before closing your previous fiscal year. Shepherd's Staff doesn't consider when a transaction was entered, just what date it was entered for, when considering what transactions belong with what fiscal year. However, if you haven't ran your year-end, you won't be able to see reports for your new fiscal year until you've ran the year-end utility.
When you run the year end utility, all income and expense accounts, at the start of a new fiscal year reset to a 0 dollar balance. This is because income and expense accounts just track the the incoming and outgoing money for just the current fiscal year. This is what makes it possible to record budgets for these types of accounts, because they revert back to 0 at the start of each fiscal year.