[dimka20]

How would one determine which species is oxidized and which is reduced w/o stochiometry?

https://imgur.com/a/KDNVB6o

Does the reduced species always gain more electrons? Normally in a galvanic cell the species with the higher reduction potential is the cathode while the species with the lower reduction potential is the anode. However, electrolytic cells are the opposite to galvanic cells, so instead of sodium (lower emf than cd) being the anode, it will be the cathode and Cd will be the anode.

[Enthalpy]

To my knowledge (not quite up-to-date), Na is still a research topic in cells. Electrolytes must be difficult. Googling "Na-Cd cell" gives this bizarre assignment as the only result - occurring identically at several places.

Which one is reduced or oxidized depends on the current direction, that is, whether the cell provides or receives electrical power. I see no relationship with stoichiometry, which is easily determined anyway, since as many electrons enter and leave the cell.

Which statements are true depends on all conventions about voltage sign, cathode and so on. Yes, reduction is an electron gain. You must have all that in your notes and books. The same metal pair can serve as galvanic or electrolytic cell, at least in education, but not every combination with the voltage sign is correct.

While voltage determinations may often be correct for both galvanic and electrolytic cells, many difficult conditions must be met to make a battery rechargeable, so changing the current direction is more something for education. If you try to recharge a Leclanché cell it doesn't work well nor many times.

Also be aware that only chemists apply correctly the historic definition of "cathode" and "anode". Electrical engineers, and even documentation for secondary batteries, give it different meanings.