Hi cjohn1,
Thank you for your question.
First of all, Exchange 2003 didnt coexisted with Exchange 2013 at the same time in organization.
1) Will there be any mail flow impact after such changes, and EOP still pointing to Exchange 2003?
A: when Exchange 2003 is coexisted with Exchange 2010 or exchange 2010 is coexisted with Exchange 2013, we suggest higher version of Exchange point to EOP.
2) Is there any issue on inter mail flow routing for 2003, 2010 and 2013?
A: Exchange 2003 didnt coexisted with Exchange 2013 at the same time in organization, so we just Exchange mail flow between Exchange 2003 and Exchange 2010 or between Exchange 2010 and Exchange 2013.
3) Any best practice or mail flow settings to be verify before such changes?
A: we could pick up message header information to check message routing by the following link:
https://support.office.com/en-us/article/View-e-mail-message-headers-cd039382-dc6e-4264-ac74-c048563d212c
In addition, we could refer to the following link migrate Exchange 2003 to Exchange 2010:
https://technet.microsoft.com/en-us/library/ff805040(v=exchg.141).aspx
We could refer to the following link to migrate Exchange 2010 to Exchange 2013:
https://technet.microsoft.com/en-us/library/jj898583(v=exchg.150).aspx
If there are any questions regarding this issue, please be free to let me know.
Best Regard,
Jim