I don't know if this is an existing option that I'm missing, or if it's a plan or something, but is there a way to get the player level (class not required) on the player's unit frame?

The only other way to acquire level is through permanently visible nameplates, and it seems that there's a current bug with the ForgeUI nameplates that requires me to manually /reloadui when I gain a level to update my visible level.

You can view your level under the Titles page of the Character panel. But considering how many people have asked/requested this, it's probably something to add as well.

Yeah, I will be adding this in v0.5

That's awesome. Is there any fix in the works for the nameplate update problem?

Nameplate update problem? If you mean slow updating of nameplates just check Fast updates under nameplates > general in main ForgeUI window