Adjust for reimbursed client expenses showing as Unbilled.

1.  If I understand you correctly, the client has billed customers for out-of-pocket expenses but not by using the Add Time, Costs, and Expense feature.  When the reimbursements were received, they were recorded as a Deposit with a "From Account" as Income.  When the original expenses were incurred, they were flagged to the customer and marked as "Billable", so now they are showing up as "Unbilled Costs".  To remove the expenses from the "Unbilled Costs" report, simply clear the "Billable" flag on the original expense entry.  You can add a memo saying "Reimbursement billed outside of QB, received on mm/dd/yy".

2.  It doesn't matter if the original expense was entered on the Expense tab or the Item tab when the expense was incurred.  What matters is that the Customer was entered and the Billable flag was set correctly.  You only need to use Items for routine expenses as a matter of convenience for data entry and for better reporting of "Item Profitability".  But costs entered on the Expense tab of a Check or Bill can still marked as Reimbursable just by specifying the customer and setting the Billable flag.