Part of this is incorrect.
Law only teleported Doflamingo when he's not in battle state or not paying attention to law, which means his haki(which is will) is not there to resist Law's DF ability. Law confirms that he can't teleport DD anytime he wishes after the Red Hawk combo.
Law teleports Doflamingo the same way he teleports Luffy sometimes, off guard, not paying Law combat attention.
In fact, it is very likely that Law couldn't shambles Vergo, which is why he had so much trouble getting his heart back, Law teleported his heart from Vergo's hand(very similar to what he did to Zeus), instead of teleporting Vergo himself. And this also explains why Vergo still believed his Haki can resist Law's DF.
Now, why did his Mountain level Amputate cut Vergo? To understand this, you have to understand how DF and Haki interact with each other in OP.
DF and Haki battles each other when used against each other(only when applied to the individual DIRECTLY). With different DF techniques, different amount of "power" is put into it, and when a character is in different state/form, he emits different level of Haki as well, for example, Shambles would have much lower DF power put into it than Mountain level Amputate, and Vergo in his full body Haki form, would emit stronger Haki than his normal form. And the one with more "power" wins.
So Mountain level Amputate "power" > Full Body CoA Vergo "resistance power" > Vergo "resistance power" > Shambles "power".
As the result, Full Body CoA Vergo gets cut, Shambles doesn't move Vergo.
Most people have this wrong, they think DF works or not is a "Haki battle" or if one individual resists ONE technique from this other individual, he resists ALL techniques, no, not the case, a DF user wouldn't battle a Haki user through Haki, he battles him through DF, it becomes DF vs Haki, NOT Haki vs Haki; also different DF techniques from the same DF ability, can have very different level of power.
Now like I said, this is only the case when DF technique is applied to the Haki user DIRECTLY(It's possible that attacking Logia user through Haki works this way as well), but if you make some type of construct through your DF, like a Gamma Knife or a Bari Bari no mi Barrier, it's a different case, the Haki user is dealing with the construct itself, instead of dealing with the "power" behind a DF technique, think of it as if Law use Takt to throw a giant boulder at you, you aren't dealing with "Takt" itself anymore, you are dealing with the boulder.
So basically Law can't teleport Vergo, Fuji, DD, BM, nor Kaido, when they are in battle state against Law.