A body's physical strength it not defined by it's heart, but by the actual body. Therefore it is not switched, as we can see when Nami (in Franky's body) wants to beat Sanji (in Nami's body) again. Nami shouldn't be able to even tickle Sanji, but if she would have hit him again in chapter 663, Nami's body might have been severely injured.
Mental strength on the other hand, does come from the heart as we can see Sanji do extraordinary things with Nami's body. Things she would physically be capable of, but not mentally. Like when Sanji blue walks his way to safety after rescuing Kin'emon's torso in chapter 672.
Devil Fruit Power
Devil Fruit Powers become part of the body after eaten, so the same can be said as with actual physical strength. For example, it was Luffy's complete body that became rubber and not his mind. Which was proven on multiple occasions. Oars couldn't use any rubber attacks on his own after he obtained Luffy's shadow. Tashigi could turn her body into smoke after her mind got switched with smoker's and Franky could still change Chopper's animal forms. With the last example, I would also like to point out that this is also a good example where mental strength is not switched, since Franky could not control Chopper's final form, while Chopper could, as seen on Fishman Island. I'm not saying Chopper is stronger than Franky perse, I'm saying that if mental strength and thus experience would have been transferred he would have been able to control the monster.
Haki obviously also gets switched, since this literally means willpower. So if the will gets switched, so will it's power. This can be seen when Sanji activaly uses Kenbunshoku Haki to locate Kin'emon's torso.
Kenbunshoku Haki is Sanji's speciality concerning Haki. When helping Kin'emon recover his torso from within the lake on Punk Hazard, Sanji actively used Kenbunshoku Haki in order to locate the samurai's torso within the dark water.
In the case of switched "souls", Devil Fruit powers are maintained within the original body, while voices and other abilities, such as Haki, go along with the switched mind/soul, as seen when Sanji used his Kenbunshoku Haki in Nami's body, due to Haki originating from the individual's spirit.
Physical heart vs Mind
It should be noted that Law only switches the mind and not their actual physical hearts. This can be seen when he switches Tashigi and Smoke's mind, while holding Smoker's physical heart in his hand in chapter 663.
Otherwhise it would have been chaos if Caesar would have put Law's heart inside himself at Dressrosa, becoming Law.