"I'm reconciling for last fiscal year, so in the case of checks written outside that period that were cashed in that period, I need to figure out how to move money between the accounts properly without changing anything that was declared on the taxes for the previous fiscal year (does that make any sense?"

Just because something cleared or didn't clear, does not apply. You don't Move anything. You issued a check, and whether or not it is cashed is part of Banking, not part of your perspective. You pay my in Nov and I don't cash it until Jan = you Incurred Subcontractor labor in Nov.

"The reimbursement was included on the employee's normal paycheck, but for some reason, the bill payment was recorded on one of the company credit cards.  Since I'm reconciling bank accounts, that transaction can't stay and I remain in balance, so how would I go about retroactively correcting this (I'm sure this isn't the only one of these I'll run into, as I've only gotten 6 months in.)"

You need a Credit Card Credit, and it gets offset to whatever is listed on the payment to the employee, because it appears your expense is entered twice using two different methods.

Or, delete the credit card  charges if it is a Duplicate entry for what was used for the payment to the employee.

Or, edit the employee payment (if this was a check and not a Paycheck), and it should not be Expense; it would be your CC account, to offset that entry error there and remove the Balance from the CC, leaving the details intact.