Let me state for the record that all time travel ends with paradoxes - this is inevitable. There is no known way to model time
travel without paradoxes (that includes the parallel universe
approach). With causality in place, time travel irrevocably will
present a paradox if you dig deep enough. Since this is fiction of
course, we just need to make it plausible enough so the problems don't
matter enough.
The Time Bomb
The simplest form of abuse is someone using this to transport an explosive device (or something who's target position will have destructive consequences) back in time, to an arbitrary location - and a time when there's no defense for the weapon.
The Grandmother Bomb
Depending on how you model your time travel rules, someone could go back and prevent the device from being made. Someone could arrive from the future (or infinite numbers of people could) to prevent it from being used. If you have time travel, anyone can have it in the future - and it lasts long enough for lots of people to come back to your time.
Too much power
Whatever the way the device works, it is obviously capable of manipulating matter at a basic level - if something goes wrong, it could just make the target space-time location explode. There's always the problem of replacing whatever is in the spot you want to put your stuff in: if you travel back in time, what if someone is in that place? What about just the air that's there? Do you replace it or push it away? I can imagine that forcing matter (or energy) to coexist forcibly with other matter, it could become accelerated to large fractions of c and just annihilate the entire place.
Prevention
If someone had such a device, it's pretty obvious that they'd want to regulate use as much as possible. To the point where they might avoid even using it themselves unless absolutely necessary. Since in your premise this is used for transportation, it will be accessible to average people.
One way of preventing the above abuses would be to intentionally engineer the device to only work with an artificial endpoint (thus, it can only send people to a specific place at a nearby time and that place has to be explicitly created for use with this device), to only be able to transport within small time distances (aka, it can transport someone from here to another planet, but when they appear, it's at most a few seconds after you start the transport) and make sure it has a time lock (like banks) and requires explicit monitoring and manual authorization for all transports.
In essence, besides having to be controlled by some kind of incorruptible authority powerful enough to prevent abuse, it would have to be engineered to prevent abuse, even if reverse engineered. It would have to be made confusing, without anyone knowing the way it works entirely and possibly break or distort itself upon any attempt at disassembly. The stakes are too high.
Of course, it can be built for a purpose and then destroyed after it is fulfilled.
Paradoxical behavior
As stated in the beginning, paradoxes are inevitable. It's easy to come up with thought experiments that end in paradoxes - it's tough to come up with ones you can test without catastrophic consequences.
I'd say that, besides testing with tiny amounts of matter and energy to evaluate causality paradoxes (using only annihilations, collisions, transmutations and conservation laws) they would have to attempt all of this within very small time frames (in the order of milliseconds at most). Unless they mess up, they should be able to figure out at least the broad rules of how causality is affected and how far-reaching the effects can be, by extrapolating from stochastic processes. They might even be capable of evaluating the validity of deterministic physical laws.
Other applications
I can't think of many - time travel is manipulating time to your advantage. You might be able to speed things up (computers that send information backwards in time to themselves have been thought of in fiction) or slow things down (like stasis in star trek). You could perhaps experiment with processes that take millions of years to complete (send something back in time, in a container and attempt to meet it in your own time - provided your device isn't locked as I proposed).
Reference frames
The only way to really achieve this would be to pick something that isn't expected to change unpredictably within the amount of time you want to travel. This can be a signature the device can detect in space-time, rather than just space (perhaps the latter would be useless anyway). Since objects don't carry a tag in the universe, it would have to detect and measure its influence on its surroundings. This means it would have to be able to predict future behaviors accurately - if it's a star for instance, it would have to include very accurate stellar and orbital mechanics predictions to make sure it can maintain the reference.
It could perhaps maintain a set of references - multiple points that it uses to produce coordinates in space-time. This would be much like how spacecraft and aviation computers are built in threes - all perform the same job and check each other for errors - if two agree, an operation is correct, if all disagree, there's an error. A reference matrix, made of multiple points, would be much more robust and would allow for some errors in the predictions.
