Also, note the amount of energy involved. At 1g accelration, each kilogram of the ship would have a maximum kinetic energy of more than 47 megatons of TNT, which is around the amount of energy you get from mixing a kilo of matter with a kilo of antimatter. (At 13g, it would be close to 620 megatons per kilogram. Strike a 1 miligram piece of space dust? Get an explosion the equivalent of 620 tons of TNT.)
There are a few reasons that doesn’t really work that well.
There are practical speed limits in space. Just because it’s vacuum doesn’t mean it’s empty. The little bit of stuff out there will become more and more of a problem as you speed up. Individual hydrogen atoms are going to be hitting with the force of hydrogen bombs at that sort of speed.
That’s, of course, assuming you have the magic drive in the first place that allows you to expend a ridiculous amount of energy to accelerate constantly like that.
I think it would still be a lot faster, as traditional way of getting to close ot the speed of light would require time for acceleration and deceleration.
Generation ships are just a napkin solution to the problem and will never be built.
As for solving it with a digital solution, that’s where nanotechnology comes in. All nanotechnology is is a physical packaging of millions and millions of assembly lines into a tiny space. These assembly lines can convergently assemble any part of the starship (and the machinery needed to bootstrap up to an advanced civilization).
So it’s basically taking all the complexity of all the factories on earth and cramming it into the space of a refrigerator. Living cells have this kind of machinery density, so we know it’s basically feasible, eventually.
Anyways, the nanoscale assembly systems would be always working as the starship coasts through space, recycling worn pieces of the starship back to gas (probably all the way down to plasma so you can separate the elements with magnets or electric fields) and then reconstructing them.
So none of the ship is more than a few years old. It remains perpetually renewed, perpetually brand new. Even the digital computers the crew ‘live’ in are constantly being recycled and replaced, with the data being copied around so that redundant copies are always kept.
So, the ship is actually always brand new, and of course there are several redundant copies of every important system. If the danger of micrometeorite impact is high enough, starships might actually travel in formations of dozens of them, with the crew as digital beings who can be freely beamed between ships, so that if any one ship hits a grain of sand and detonates it’s antimatter tanks, the mission can continue and none of the crew are lost. (the spacing is a light second or so)
Your body. Human beings are physical creatures. Non-physical creatures are ideas. Data is not a human.
Presumably, the humans on board would be up for doing some maintenance. I don’t think it’s that much of a stretch that the human crew could maintain a spaceship for hundreds or even thousands of years. Certainly, the reality is if we ever do decide to go to another star we are looking at hundreds of years unless some of the more radical technology envisioned becomes a reality.
I assume someone has already mentioned to the OP that fiber optics use light at light speed (in the medium of the fiber optics of course) and that radio waves also propagate at the speed of light in whatever medium they are in…and that both are perfectly able to transmit information.
As for faster than light, well, I’m sticking with entangled articles eventually being able to be used to transmit data at faster than light speeds, if anything can. Or maybe small wormholes in the quantum foam. If not, then definitely magical unicorn power!
Playing around with this idea, there is a theoretical maximum amount of information contained in a given volume. On a human scale, one gets something like 10[sup]42[/sup] or 10[sup]43[/sup] bits.
In theory, humans can maintain a generation shop for hundreds of years. But how much technology and other stuff is needed to do so? For example, let’s say you use LED bulbs for lighting (particularly for growing food). These bulbs might last 50,000 hours, or about five years of continuous use. So for a journey of a hundred years, you’ll need twenty replacement bulbs for every one on board. Are you going to keep that many spares in inventory? Are you going to manufacture them en route? You’re going need a series of complex factories to make all of the different things that will need to be replaced during the journey.
That’s not really correct. The “Ship of Theseus” thing is just pure abstract philosophical rumination. The silicon chip replacement thought experiment was proposed in support of the computational theory of mind, an important if still somewhat controversial premise in cognitive science. The two things are only very superficially related.
As for the OP, I see no difficulties here. Teleportation of physical objects at a macroscopic scale and maybe even humans may be possible at some point in the future, and would pose no difficulties to relativistic physics. You just simply cannot travel or transmit information faster than light because of the way that space and time are inextricably bound together.
Nitpick back at ya. You don’t need to set light to the torch - just wave it - ambient light bouncing off it will transmit information about its movement at the speed of light. 
Then the calculator is missing a parameter or two. According to the calculations in SR, as you approach c, your mass increases, which means the force needed to accelerate you increases: a thing with non-zero rest mass acquires infinite mass upon reaching c, which means it requires infinite energy to accelerate to c.
Then there is the time dilation thing, that causes the 10000G acceleration to curve downward.
10000g getting you to Alpha Centauri in a day is about right (including turning around halfway and braking). Unless I missed a joke.
Without checking the calculations, it’s only a day inside the ship. Outside the vessel it would be slightly more years than the number of light-years of distance traveled.
No, your mass does not increase, only your energy. And while it’s impossible to reach c, it’s possible to have a constant, continual acceleration indefinitely, if you have enough energy available. The calculation is correct.
As an aside: The Queen song “In the year of '39” is about a space journey where the travelers spend a year in travel, and return to Earth a hundred years later. It’s usually interpreted as being about relativistic time dilation, but to get that degree of time dilation over that time span, the ship has to be under about 800 gees the whole way (it’s a surprisingly easy calculation).
To someone outside the ship (and not moving in respect to the starting inertial frame), it would seem as though your acceleration tapers off relatively quickly, by the time of flip-over, the observer would detect your acceleration as being tiny. (speaking of which, flip over would probably actually take a long time in the non-accelerated frame, it’s always considered to be instantaneous in the calculations, but at these speeds, you’d probably cross most of a light year just on the maneuver, keep that in mind while planning your next interstellar trip) You are experiencing 10000gs, or 100000m/s/s (rounded), and your gamma is stupid, so the non-accelerated observe would measure your acceleration as .000~01 g’s (not sure how many 0’s exactly, but you get the point), and your ship would probably be a bit on the short side too. So, to the outside observer, it does seem as though it requires more and more thrust to keep accelerating, or rather, that you are accelerating with less and less thrust. But, to the passenger on the ship, the acceleration is decreased precisely with the time dilation, so it seems as though the acceleration is the same the whole time(because it is, minus flip-over).
Your outside observer could naively chalk that up to you having more mass to push, but that’s because the term “relativistic mass” is misunderstood, and should probably have never been named that, as it is misleading. It’s another term in the f=ma equation, the proper equation is f=p(actually a greek symbol) x m x a. In every experience you have ever had, that “p” has never been far enough from 1 to make any difference, but it’s there. It doesn’t get added to the mass, it is its own entry in the equation.
And you would think Brian May wouldn’t have made that blunder. 
In fact in special relativity force is not proportional to acceleration at all, so the acceleration produced by a force is not necessarily along the same axis as the force.