Is there a legally mandated filing fee and/or deposit in the last instance?

Electoral Act 2006, 105(3)

The High Court may, onapplication made by any person to whom any costs, charges or expenses arepayable under this Act, order the same to be paid out of any deposit made tosecure the same, or by any surety who gave a recognizance to secure the same:

http://aceproject.org/regions-en/eisa/ZM/Electoral%20%20Act%202006.pdf

EU 2011 EOM Report, pg 20

A person bringing an election petitionin the High Court must pay €22 to court as security for costs.

http://www.eueom.eu/files/pressreleases/english/eueom_zambia_final_report_en.pdf