I won't type a huge answer with a bunch of math because other people have already done that, and the math checks out. What I will do is give you two points that will help get the gears turning in a more creative way, plus one means of detection that I don't think anyone has mentioned yet!
1. Regarding the one week deadline from detection to arrival: the distance between Pluto and Earth is 4.6 light-hours, which simply means that travelling exactly at the speed of light it would take them 4.6 hours to make it to Earth. Since they are having tech problems, and they're starting slightly beyond Pluto, you can basically adjust these numbers however you want to make it work out for the story. Example: if 1.00c = 4.6hr travel time, then multiplying the travel time by roughly 36x would give you a travel time of 168 hours, or almost a week. That would be (1/36)c, or 0.027777c, or 2.7% the speed of light, which is completely believable for a species capable of FTL travel but currently experiencing some technical difficulties! Feel free to customize this based on your needs, or even do something like require the ship to slowly accelerate and build up speed, which changes the equation. I didn't consider this until now, but if the typical FTL drive is broken on the ship, they will need to resort to conventional means of acceleration, which will result in G-Forces being exerted on the crew. They may not be evolved for that, and it won't be an insignificant amount of G-forces either; it would probably kill humans to accelerate like that. I don't know exactly what the Gforces would be, but there are some physics calculators online that are based around F=ma, and will tell you the acceleration required to reach a certain speed in a certain amount of time, and then you can look a that acceleration and compare it to the acceleration of Earth's own gravity (9.8 m/s^2) which is one "g" of acceleration, to see how strong it would really be. They'll need to use some kind of tech that we don't have yet to cancel these g-forces, like "inertial dampeners" which have been a staple in science fiction for decades but don't have much basis in reality, or maybe a shield that provides a different "frame of reference" inside the bubble, so from the ship's perspective and the perspective of the crew inside the ship, it doesn't feel like you're moving at all because the shield bubble is like a bubble of "still" space, but the bubble can accelerate through our space as fast as it wants to, or as fast as their malfunctions allow them to. 947
This is similar to how the Alcubierre Drive would work, except it would only be a bubble of local space around the ship and wouldn't have any type of physical shielding capabilities.
NOTE: The information being received by scientists/military on the ground will be 4.6 hours out of date due to the distance and light delay. This means that when the scientists first spot the ship and make the prediction that the ship will reach Earth in roughly 168 hours, the ship actually appeared 4.6 hours ago and began moving then, giving it a 4.6 hour head start. The scientists should be aware of this, as physicists have known about light delay for hundreds of years and have been dealing with it personally since the dawn of radio communication. Also, as the alien ship gets closer and closer, the light delay will shrink and shrink until it gets to 0 (light delay from Earth to the Moon is about 1.3 seconds for reference) so it isn't like the alien ship will just appear on Earth when it looks like it is still 4.6 hours away.
1. Regarding the detection: we currently have some pretty new devices called gravitational wave detectors that we are using to gather information about fast rotating black holes and pulsars. You're right there would be a "BANG!" when the ships jumped back to local relativity, but it wouldn't be a sound wave since space is a vacuum. While there could certainly be a blast of radiation released when the ship drops out of FTL speed (as is theorized would be released by an Alcubierre Drive, which is one of the current ideas for a true FTL drive) there will also be "gravitational ripples" released by the sudden appearance of large ships. Usually, there are NO such ripples ever caused by matter appearing or disappearing suddenly, other than at the quantum scale. Even when a star explodes, nearly 99.9% of all the matter is still there, it's just expanding now. Similarly, when matter is sucked into a black hole, all the matter is still there contained inside the black hole: the mass of the singularity just rises based on the mass it is consuming. However, for matter to just appear seemingly out of nowhere? 947