We could probably moving planets with today's (PL 5) tech, provided you can endure to wait centuries (or more) to see even te smallest progress. It's just a matter of building thrusters on suitable spots and then firing them every time they point in the right direction; what makes it utterly impratical is the amount of effort and time involved.
It doesn't take PL9 to move planets around; it just takes massive amounts of resources.
At PL6, with fusion torches and the ability to collect resources from a whole system, it is probably feasible to significantly alter the orbit of an Earth-sized planet in a matter of several decades.
At PL7 we could rely on much less intrusive gravity induction engines, much more efficient anc compact power sources and the resources of multiple star systems (thanks to FTL travel); a big interstellar nation might even gather enough resources to build a giant, spaceborne tractor beam facility with enough power to move planets around (maybe in a matter of weeks, if the power isn't that much). A ship able to incapsulate a planet and bring it around over interstellar distances is probably too much of a strain.
It could even be possible to create artificial planets that
are starships, like the Eldar Craftworlds in Warhammer 40,000. A solution might be to drop self-replicating robots (or nanobots) programmed to turn an existing planet into a giant ship.
At PL8 resources could be gathered from an entire galaxy and a galaxy-spanning nation might already keep around several FTL-able "planet pushers" for terraforming and military purposes. Planet-carrying ships would be still great marvels.
Craftworlds become more of a possibility.
At PL9+ anybody with a car-sized device who sprouts enough technobabble can move planets around by reducing them the size of marbles and carrying them in their pockets. People wonder what business you have with planets when everybody can have his own pocket universe;and if you really move a planet around, the Bad Guy will blow it up with a single shot of the latest superweapon just for spite.
A planet-sized jumpgate along the orbit would be feasible at any PL, but it might require far more resources that it's worth, especially if it can't be moved to other systems at a later point.
As for moving planets safely:
1) Anything that has to deal with inertia and Newtonian action-reaction is going to make a big mess, unless the process is extremely slow.
2) Anything that changes (even just temporarily) the axial tilt, rotation speed or distance from the sun of the planet will catastrophically blow up weather patterns; the
twister movie would be an example of sunny and calm weather, by comparison. The only solution is to make the whole planet an artificially regulated environment sealed from outside sources.
3) Anything that exposes the planet to strong tidal forces will make the surface a very bad place to live.