Is there a legally mandated filing fee and/or deposit?

Turkmenistan
Answer
No
Source
Избирательный Кодекс Туркменистана (2013-05-04)
Parahat.info (Information portal on Tajikistan),
Избирательный Кодекс Туркменистана (2013-05-04)
Parahat.info (Information portal on Tajikistan),