A rules question regarding Bard Mercurial: Can I exhaust Legend cards even if they don't have an explicit exhaust ability? If yes, do the passive effects of Legend cards like Ahri or Leona still remain active while they are exhausted?

Yes — Bard, Mercurial lets you exhaust your Legend as an additional cost to play him, even if that Legend has no exhaust ability of its own, and the Legend's abilities (passive or triggered) stay fully active while exhausted.

Exhausting is an action that marks any non-spell Game Object on the board as spent (Rule 414.1), and Legends are Game Objects on the board, so your Legend is a legal target for Bard's cost. You don't get to exhaust your Legend for free just because you want to — exhaust is a Limited Action normally performed only as a cost or when an effect tells you to — but Bard's ability explicitly provides that instruction. One caveat: if your Legend is already exhausted, you can't pay the cost, since a Game Object that's already exhausted can't be exhausted again (Rules 414.1.b, 414.4).

Exhaustion only prevents activated abilities that require exhausting as a cost. It does not turn off passive or triggered abilities (see FAQ 604). Ahri's and Leona's Legends are both triggered abilities ("When an enemy unit attacks..." / "When you stun...") with no exhaust requirement, so they function normally even while your Legend is tapped (FAQ 4574).