Invoices or any transaction in Receivables are removed from Current and sent to History when the Current Amount of the transaction is 0.00 and the appropriate 'clear to history' function is run. That function could be either Printing Statements or running Period End, depending on what you have defined in your company profile (either Billing Cycle or Period End, respectively).

As mentioned in the previous post, transactions with a Current Amount not equal to zero will not be cleared to history. For your invoices dated 01 Jan 02, what are the current amounts? You can print the Customer Transaction report or view a Customer and look at the Transactions screen. If not zero, then this is the answer. You need to perform steps such as mentioned above to get the amounts to zero and allow clearing.

If the current amounts of these invoices are zero, then you need to run Period End / Print Statements to purge. In the mean-time, you could select the option for 'Outstanding Transactions only' when printing statements, in order to not print completed invoices with zero current amount.

If you are still having problems with this, please post an example of an invoice that you see printing on a statement with this date, the current amount of this invoice, and the method of clearing transactions to history.
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As for your other question regarding printing invoices, do you have a logo inserted into the header section? It could be that your printer is out of memory when printing that specific document and thus prints blank instead of the logo. Perhaps you have to send few invoices at a time to the printer, to see if that helps.