Does a relativistic rocket's combustion slow with time dilation? If a relativistic rocket travels slower in time, does that mean the actual combustion in the thrusters happens slower and slower from an outside frame of reference? Is this part of what causes a relativistic rocket to lose acceleration from an outside frame of reference? Does this mean that an unfueled ship, such as one with a laser sail or magnetic sail, would be faster, as the propulsion would not happen inside the ship, and thus, not slow with time dilation?
 A: Yes, a vehicle with some constant power in it's rest frame would appear to have lower power from other frames.

Is this part of what causes a relativistic rocket to lose acceleration from an outside frame of reference?

The mechanism doesn't seem to matter to me.  If I create a hypothetical vehicle with constant proper acceleration, other observers will not see constant acceleration.  It's not necessary to attribute it to slower combustion or anything else.  So I would say no, it's not a separate issue.

Does this mean that an unfueled ship, such as one with a laser sail or magnetic sail, would be faster

Now you have a vehicle interacting with something else.  As the vehicle picks up speed, it will be moving quickly with respect to that external energy source.  It will become more and more difficult to deliver the energy.  The laser on your planet  delivers constant power on the planet's frame, but progressively less power in the rocket's frame.
A: 
If a relativistic rocket travels slower in time, does that mean the actual combustion in the thrusters happens slower and slower from an outside frame of reference? Is this part of what causes a relativistic rocket to lose acceleration from an outside frame of reference?

Of course. Slow motor does its job slowly. The job of  rocket motor is to cause a velocity change delta v to occur, preferably swiftly . $$a= \Delta v  / \Delta t $$

Does this mean that an unfueled ship, such as one with a laser sail or magnetic sail, would be faster, as the propulsion would not happen inside the ship, and thus, not slow with time dilation?

Hmmm, If there are lasers floating in space, and those lasers give a spaceship a short push when the ship passes, then the energy that is stored in the lasers gets transferred to the ship in a non-slowed down manner.
A: It doesn’t matter if the energy that propels the spaceship is from an external source or an internal source. The transfer of momentum between the photon beam either hitting the “sail” or the photon beam leaving the ship’s photonic engine will decrease as the ship approaches light speed because the light will be further and further red shifted until from the ship’s point of view due to the Doppler effect until thr wavelength appears so spread out it is essentially black and can deliver no further energy. If the ship COULD travel at the speed of light, the light from the lasers would never reach the sail and the photons from the engine would never be able to exit the engine. Like at the event horizon of a black hole time would appear to stop and the ship would appear to blink out of existence because any light emitted from the ship would never reach the distant stationary observer just as light from distant parts of the universe moving away from us faster than the speed light will never reach us.
