If the universe is 15 billion years old, how can it be larger than 15 billion light years across?
Well, the answer is general relativity. I take it that you're starting out in SR. Special relativity is "special," because it is a special case of GR, namely that there are no gravitational fields or accelerations, and that space is flat and static. Only in that regime is it strictly true that relative velocities must always be less than c. GR allows any velocity you like, including v>c, so long as the two observers cannot agree on a common Lorentz frame (an inertial frame in which special relativity applies). This can happen in strong gravitational potentials, or in, say, an expanding universe.
In English, the two observers cannot give a definition of "inertial" that works in both places. They are separated by curved or expanding spacetime, which is not well-described by either observer's inertial frames.
In the example of the universe, for instance, objects recede from you not because they are in motion, but because space is being created between you and the object. The universe can create as much space as it likes, even so much that it appears that distant objects are travelling faster than c. So long as objects don't reach c relative to any local observer, to whom the expansion is negligible, this is perfectly okay with Einstein.
So yes, objects in the universe can travel faster than c away from us due to the expansion of the universe, and the universe itself can be much larger than the age horizon. We are never able to see anything further than 15 billion light-years away, of course.
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