What are the best stories where a hero temporarily battles a villain belonging to another character?
Strangely, in the early days of the Avengers before they truly had an established cast of villains (Ultron, amazingly, didn't show up until the fifth or sixth year or so) they did this story quite a bit. AVENGERS #25 had them battle Doctor Doom, who used a rumor to lure them into Latveria, where he used a gigantic plexiglas bubble that surrounded the entire COUNTRY! (Even by Silver Age standards, that's a grandiose piece of equipment).
One of the best stories of this type would have to be the Mad Thinker vs. the Avengers in AVENGERS #30, especially because of the incredible art by Don Heck; the Thinker here had perfectly normal hair, but had EYES - oh, what EYES - that were dark and totally evil, to see how rotten to the core he is. The Thinker assumed a Rodan-like pose while coming up with plans. The Thinker had a logical motivation to attack the Avengers: he wanted to steal Tony Stark's electronic secrets. The reason this story stands out is that the Mad Thinker with his supreme tactical ability, WON the day...he only lost because of Hercules's appearance, a technicality.
Another would be Magneto's battle against the Avengers in SUPER-VILLAIN TEAM-UP, where he defeats Yellowjacket and Wasp in a profoundly novel way, by magnetizing iron dust around them, and changing it into spheres, starving them of air.
Over at DC, one of the better such "villain fighting a different hero" stories is in DETECTIVE COMICS #353 (1966) “The Weather-Wizard’s Triple-Treasure Thefts” by Gardner Fox back when trading foes was infinitely less common. The premise had Weather Wizard trading his digs in Central City for Gotham.