If a muon travelling fast can “extend” its lifespan due to relativistic effects, would the muon see itself travelling faster than light? In other words, if I put a bomb set for 2.2 microseconds and send it out at .99c, would it travel further than (.99c x 2.2 microseconds)? And if it does, like muons do, wouldn’t the bomb be able to tell that it has gone faster than light for the 2 microseconds it was alive?
 A: In the frame of the bomb, you would travel away at 0.99 c for a distance of (0.99 c * 2.2 microseconds) before it exploded.
So, since in its frame you travelled away at 0.99 c from it, it would know, that in your frame, it travelled away at 0.99 c from you. In no frame, would it travel faster than c.
A: Yes, it would travel further than $0.99c \times 2.2 \,\mu\text{s} $. It would travel $\beta \gamma c \times \Delta t$ because of time dilation. This does not mean however, that the muon or the bomb see the surroundings moving faster than the speed of light. This is because in the frame of the muon distances are relativistically contracted. So if you have some rod that you use to measure the distance the muon has passed, then the muon sees this rod contracted, so that if the muon decayed at the end of the rod, it would still find $\beta = 0.99$.
A: 
if I put a bomb set for 2.2 microseconds and send it out at .99c,
would it travel further than (.99c x 2.2 microseconds)?

It depends on which frame of reference you are considering. In the lab frame, it would indeed travel further than that.
In its own frame, it would stay at rest while the lab would travel away to a distance of .99c x 2.2 microseconds before it explodes.
In the frame of the bomb, it stays alive for 2 microseconds, and within that duration of time, the earth, the lab etc. would travel away from it at .99c . So, it would not say that it has gone faster than light for the  microseconds that it was alive.
A: Time dilation goes along with length contraction.
Consider a muon that we measure travelling at 50 km when it "should" have only travelled 600 m. We say it travelled 50 km because we have conventional reference frames that are convenient for us to use that say that distance is 50 km. The muon, however, is moving a substantial fraction of the speed of light relative to our reference frames, so lengths along its direction of travel are contracted in its own frame of reference (relative to ours).
In the muon's frame of reference, it only survived 2.2 microseconds, and travelled 600 m. In our frame of reference, both the time it survived and the distance it covered are different.1 The equations of relativity tell us how measurements of both time and length taken in different reference frames should correspond, and work out such that in no frame of reference (no matter how arbitrarily constructed) does measured distance divided by measured time exceed the speed of light.
When you ask "wouldn’t the bomb be able to tell that it has gone faster than light for the 2 microseconds it was alive?" you're mixing up frames. You're talking about the time measured in the muon/bomb's own reference frame but the distance measured in our reference frame (or at least, one that the bomb is moving 0.99 c relative to). If you measure the time and the distance in the bomb's frame its speed will be less than c, and if you measure the time and the distance in our frame the bomb's speed will be less than c.

1 The start and endpoint are the same in both frames though; it's the distance measured between those two points that is different, not whether or not the muon actually reached a given position.
A: 
would it travel further than (.99c x 2.2 microseconds)?

If you send it out at a velocity, relative to Earth, of .99c, then Earth's spatial component (that is, the spatial component in Earth's reference frame) of the displacement between it being sent out and its exploding is more that .99c * 2.2 microseconds.
The bomb's spatial component of the displacement will be zero; in the bomb's reference frame, it is stationary. It will view Earth as moving in the opposite direction, and the distance it observes Earth moving will be .99c * 2.2 microseconds.
Now, that last statement is rather complicated, as the distance it observes Earth moving, by Earth's yardsticks, will be more than .99c * 2.2 microseconds. That is, if someone on Earth were to put out a ruler and put tick marks such that the tick marks are, in Earth's reference frame, each 1m apart, then the bomb will observe the ruler moving at .99c, and the tick mark next to the bomb when it explodes will be more than the number of meters that the calculation .99c * 2.2 microseconds gives (that is, .99c * 2.2 microseconds = 650 m, but the bomb will see more than 650 tick marks fly by). This is because in the bomb's reference frame, the ruler is traveling near the speed of light and thus contracted, and so the tick marks are less than 1m from each other.
