Does a interstellar spacecraft traveling at relativistic velocity require continous thrust to maintain velocity? Assuming completely empty space, does a spacecraft traveling at 0.5 C require continuous thrust to avoid deceleration?
If the spacecraft is traveling at 0.5 C, does it's relativistic mass act upon objects that it passes by?
It is understood that space is not truly empty.  The question was designed to consider only relativistic effects.
On a continuing thought however, If the spacecraft is traveling at 0.5 C and starts to decelerate, does it need to decelerate it's relativistic mass or it's rest mass?
 A: In completely empty space a spacecraft travelling at any speed does not need thrust to stay at that speed. It only needs thrust to accelerate or decelerate. Space isn't empty, so a spececraft would need a small amount of thrust to maintain it's velocity, but in most circumstances the thrust needed is tiny. This issue was discussed in Would a fast inter-stellar spaceship benefit from an aerodynamic shape?.
No matter how fast a spacecraft travels it doesn't develop a stronger gravitational field. This is because the source of the field is the stress-energy tensor not just the mass. For more discussion of this see Relativistic mass as seen by different observers.
A: First, a quibble:  in completely empty space, the speed of a spacecraft is ill defined.  What would it be moving relative to?  Empty space?
Now, let's say that the spacecraft is moving relative to the CMB at 0.5 c in otherwise empty space.  I'd like to say that no thrust is required because there is no drag on the spacecraft to decelerate it.  However, due to the blue shift of the CMB in the direction of travel, I suspect that there may actually be some radiation pressure decelerating the spacecraft.
Update:
Another minor quibble:  Mass acting on other masses is a Newtonian conception of gravity.  In the context of GTR, mass does not "act upon objects"; mass (energy) "acts" on (curves) spacetime.  This curvature is related to the density and flux of energy and momentum which are frame dependent quantities (though they are expressed in a frame independent way via the stress-energy tensor).
The bottom line, as Ron points out, is that, in the coordinate system in which the rocket moves at 0.5 c, it is the relativistic energy and momentum of the spacecraft that relate to the spacetime curvature as "seen" in that coordinate system.   
A: Nope.  In fact the principle of inertia implies that once something get's going in empty space it will keep going at that speed forever. Although for your second question yes the presence of other bodies would accelerate you toward them since you would follow a geodesic (following a geodesic being the appropriate generalization of the principle inertia).  The speed doesn't really matter (although obviously it changes the degree to which you are pulled and if you are moving fast enough you can escape the pull of the planet but NOT if you don't accelerate).  So no in COMPLETELY empty space you would not need continuous thrust (or any additional thrust) but if there are other bodies around you will need a bit of thrust to avoid them but if they are far away this thrust could be somewhat negligible.
Edit: (As was pointed out by another answer if you were moving in empty space you wouldn't even "know" you're moving but I think it's still a reasonable question if space is "almost empty" like if you're between the Andromeda and the Milky Way (pretending nothing else is floating between them). Then you could "see that you're moving" but the bodies wouldn't require you to really thrust at all to keep going along a constant velocity (although strictly speaking you'd be following a geodesic with a minute curvature).
A: I've nothing to add to the kinematics and relativity of the situation, which others have properly addressed.  But if you're interested in the topic of interstellar travel more generally, this article by a former NASA scientist is great reading:
http://arxiv.org/abs/1101.1066
Abstract below:

Energy, incessant obsolescence, and the first interstellar missions
Marc G. Millis
Projections for the earliest interstellar mission possibilities are calculated based on 27 years of data on historic energy trends, societal priorities, required mission energy, and the implications of the Incessant Obsolescence Postulate (Where newer probes pass prior probes). Two sample missions are considered: launching a minimal colony ship where destination is irrelevant, and sending a minimal probe to Alpha Centauri with a 75 year mission duration. The colony ship is assumed to have a mass of 10^7 kg, and the probe 10^4 kg.
It is found that the earliest interstellar missions could not begin
for roughly another 2 centuries, or 1 century at best. Even when
considering only the kinetic energy of the vehicles without any regard
for propellant, the colony ship cannot launch until around the year
2200, and the probe cannot launch until around 2500. Examination of
the Incessant Obsolesce Postulate shows that it becomes irrelevant
under several conditions.

A: For the drag caused by the CMB see http://prao.aps.org/story/v12/st22 and the references there.
