Of course, this is assuming that we can build fast interstellar transit systems...which is a huge if.

A good paper about using such lensing: "Interstellar radio links enhanced by exploiting the Sun as a Gravitational Lens" by Claudio Maccone.

Same reason we aren't doing a lot of other things - nobody wants to pay and sending an interstellar probe would be extremely expensive and full of engineering difficulties. It probably makes terraforming of Mars look quite feasible.

Agreed. Don't send a man to do a machine's job, and this is definitely a machine's job.

In the year 2200, if a 10 year old child were to travel to the planet at a speed of 0.9999c, then were to immediately turn around and travel back to Earth, they would be approximately 50 years old. However, what year would they arrive back on Earth?

Assumptions:

- no acceleration or deceleration time.

- the planet is exactly 20 light years away.

- the velocity of the ship remains exactly 0.9999c while in transport.

(The answer is not 2240. It's much greater. My question is, how much greater?)

The child is traveling 40 light years at 0.9999c, which takes about 40 years + 35 hours. However the child won't arrive back 50, the child will not yet be 11. See http://www.wolframalpha.com/input/?i=time+dilation+traveling... for the exact age.

See http://en.wikipedia.org/wiki/Twin_paradox for more on this, including an explanation of why your belief about the age of the traveler is wrong.

If the child were moving at 0.9999c (which is less than 1.0000c) and the planet is 20 light-years away, then how could it take less than 20 years for the child to reach it? Let alone ~0.5 years?

EDIT: Here's an explanation from a friend:

  light always travels at c even if you're already moving close to c
  but it's impossible for anything to travel faster than c
  so if you're traveling at .9999c
  the passage of time must be scaled for the traveler
  to make light on the ship appear to move at c
  even though it's only moving at 1-.9999c

  If the planet is 20 light years away, you're saying we 
  could reach it in less than a year if we attain a velocity
  of 0.9999c?

This is also why particles traveling at c cannot possibly decay: no time passes for them. A photon is everywhere at once, from its own point if view.

I also hope to have an understanding of what it means for a photon to be everywhere at once, from its own point of view.

In the end, I guess I just want to understand the universe just a bit more than I do.

  This is also why particles traveling at c cannot possibly
  decay: no time passes for them. A photon is everywhere at
  once, from its own point if view.

Furthermore, all particles (electrons, protons, quarks, etc.) have wave-particle duality just as much as photons do.

And rounding it out, different observers disagree on which events are simultaneous. In particular until the traveler turns around, the traveler thinks that the Earth was left recently. After the traveler turns around, in the new reference frame the traveler left the Earth close to 40 years prior.

Sci-fi dealing with this topic: Forever War, Ender's series, a few stories from Niven's Known Space universe.

Here's a really mind-blowing thing: If you were able to accelerate at a fairly reasonable rate indefinitely - say - using an interstellar ramjet, you could conceivably circumnavigate the entire universe within a human lifetime (in your own frame of reference, of course)

Of course, interstellar ramjets might not actually work in practice... but still, the concept of time/space dilation holds.

2828 years + 2200 = 5028 A.D. says wolfram alpha

  40 years at 0.90000c =   91.76 years
  40 years at 0.99000c =  283.55 years
  40 years at 0.99900c =  894.65 years
  40 years at 0.99990c = 2828.00 years
  40 years at 0.99999c = 8944.00 years