[RPG] Are there “free actions” in D&D 5e

RAW: No

As you mentioned, the description of Nauseated clearly states that you can only take move actions, and a down-conversion of move->swift is not possible by RAW.

The only action such a character can take is a single move action per turn.

Take a look at the very similar Staggered condition, where free/swift/immediate are explicitly allowed, implying that needs to be the case.

A staggered creature may take a single move action or standard action each round (but not both, nor can he take full-round actions). A staggered creature can still take free, swift, and immediate actions. A creature with nonlethal damage exactly equal to its current hit points gains the staggered condition.

(emphasis mine)

RAI: Maybe?

People in this paizo thread point out that there are some class features and spells that imply that swift actions should be possible.

• The Persistence Inqusition has this ability:

  Inner Strength (Su): At 6th level, once per day, you may heal yourself as a swift action, healing 1d6 hit points for every two inquisitor levels you possess. When you use this ability, you can also remove one of the following conditions from yourself: blinded, confused, dazzled, deafened, frightened, nauseated, shaken, sickened, or staggered.

• The "Range: Personal" spell cleanse:

  This spell [...] ends any and all of the following adverse conditions affecting you: ability damage, blinded, confused, dazzled, deafened, diseased, exhausted, fatigued, nauseated, poisoned, and sickened.[...]

Personally, I find these arguments very weak. Neither of these abilities are specifically designed to counter Nausea, both are rather cleansing a broad variety of status effects. The spell argument is further compromised by the fact that casting spells is also explicitly forbidden while Nauseated, which definitely points to developer oversight, which are not uncommon[citation needed] with such an expansive ruleset. By extension, I guess the author of Inner Strength didn't expect the Persistence Inquisition not to work correctly.

On page 67 they are listed at #9 on the infernal hierarchy in the ranks of the greater devils. This is part of the general section on the Infernal Hierarchy.

Along with this text also on page 67.

Greater Devils. The greater devils include the pit fiends, erinyes, horned devils, and ice devils that command lesser devils and attend the archdevils.

So regardless of their origins the designer's intent is that they are part of the Infernal Hierarchy.

And note the exact text on page 69 is

Legends tell that the first erinyes were angels that fell from the Upper Planes because of temptation or misdeed

So at first the erinyes were fallen angels and probably the oldest and most powerful still are. But this implies are also erinyes that were promoted through the ranks.