2) The wormholes would stay at the position they were initially created
That wouldn't make sense: it's meaningless given the principle of relativity. All sublight velocities are equivalent, and every place in spacetime at a later time is a different place. No velocity keeps you in the same place any more than any other.
Also, regardless of what other forces may affect the mouths, gravity is universal, so nothing could prevent you from gravitationally "towing" the mouths to whatever position and velocity you wanted. (As for other forces, if the mouth is stabilized by exotic matter inside some sort of conventional-matter containment structure, you might be able to just push it.)
1) The wormholes would travel with the planets, constantly changing the spacetime-wrapping.
The motion of the mouths in spacetime (the ordinary part of spacetime, I mean) has nothing to do with the shape of the wormhole region connecting the mouths. The behavior of the wormhole depends entirely on the nature of the magic that created and maintains it, but I don't see how it could even notice careful gravitational towing, given the equivalence principle.
(I suppose it's not impossible that our 4D spacetime could be a surface in a higher-dimensional spacetime with an extrinsic shape and extra dynamics, in which case the mouths might have a preferred position and speed, and trying to move them might actually stretch the wormhole. But that would be a dramatic alteration of general relativity and I'd expect it to show up in some way in cosmological data even if there aren't any wormholes.)
If option 1 was the case, what would happen if someone entered the wormhole? Would he keep the velocity of the planet where he entered at the planet he came out, which means he would move away from the target planet? Or would he adopt the velocity of the target-planet?
If you freefall through the wormhole, you'll come out with the velocity you went in with, transported along your path through the wormhole. In the simplest case, the wormhole would be something like the portals in the game Portal, and your speed relative to the exit mouth would equal your speed relative to the entrance mouth. (Also, much as in the game, you could likely accelerate to arbitrary speed by repeatedly passing through the wormhole if the mouths were appropriately arranged. This is just another reason that wormholes seem unlikely to exist in reality.)
Note that "the velocity of the planet where he entered at the planet he came out" is ambiguous, since there's more than one way to transport velocity from one place to another. If the mouths are in relative motion in the non-wormhole part of spacetime, and you enter one mouth at a low speed, you'll come out at a low speed relative to the exit mouth (velocity transported through the wormhole), not at a low speed relative to the entrance mouth (velocity transported the long way around).