Wrong. A sample of 1 (the fact that you are in this era), can give you a surprisingly large amount of information about the sample space (people born, sorted by era). The chance of your sample of 1 being in the first 10^-50 of the sample space is pretty small. See also: the German tank problem
Does not apply. The Allies witnessed more than one tank, from which they were able to infer a sequential numbering scheme, and thus derive all sorts of information about German manufacturing capability and total armaments.
What if the Allies' statisticians just had one representative tank sample, with the number "12" stamped on it. What could they infer then? Nothing. Maybe it's a sequential serial number and they should infer that Germany only has ~24 tanks. Or maybe they should take as a prior that Hitler would send his older tanks into battle first, in which case you'd expect an early serial number and who knows how many tanks there are. Maybe it's a production number and there are 12 different models of this tank class and unknown production runs of each. Maybe "12" just means it was made in December, or from factory #12.
Ok, some of these examples are specific to the tank problem and don't generalize to anthropic reasoning, but the point still stands: it is bad epistemology to generalize from one example. The only definitively valid anthropic conclusions is that there is at least one representative sample, not zero (i.e., that we live in a universe and at a specific time where human beings do exist). For the doomsday hypothesis, that tells us nothing.
(Another form of the anthropic principle applies to the Fermi paradox, but in this case we can infer other sample points from not seeing evidence of extraterrestrial intelligence in our local neighborhood. This is a very different line of reasoning, and does not apply here as far as I can tell.)
EDIT: To be clear, the specific point where the German tank problem analogy falls apart, is in its underlying assumption: that the tank (us) was selected at random from a pool of armaments (possible birthdays) with a uniform distribution. This is a completely unwarranted assumption. If, on the other hand, reincarnation were real and timeless, then you could look at "past" (future?) lives and start doing statistical analysis. But we don't have that luxury, alas.
One data point cannot tell you literally nothing. If it did, then by induction, any finite number of data points would also tell you literally nothing. In most cases, a single data point tells you very little, because it is somewhat rare for people to be considering different hypotheses in which a single observed data point is orders of magnitude more likely in one hypothesis than in the other. The doomsday argument is an exception: It is 10^10 times as likely for a randomly selected human to be one of the first tenth of humans who have ever lived than it ...
Ideas developed with Paul Almond, who kept on flogging a dead horse until it started showing signs of life again.
Doomsday, SSA and SIA
Imagine there's a giant box filled with people, and clearly labelled (inside and out) "(year of some people's lord) 2013". There's another giant box somewhere else in space-time, labelled "2014". You happen to be currently in the 2013 box.
Then the self-sampling assumption (SSA) produces the doomsday argument. It works approximately like this: SSA has a preference for universe with smaller numbers of observers (since it's more likely that you're one-in-a-hundred than one-in-a-billion). Therefore we expect that the number of observers in 2014 is smaller than we would otherwise "objectively" believe: the likelihood of doomsday is higher than we thought.
What about the self-indication assumption (SIA) - that makes the doomsday argument go away, right? Not at all! SIA has no effect on the number of observers expected in the 2014, but increases the expected number of observers in 2013. Thus we still expect that the number of observers in 2014 to be lower than we otherwise thought. There's an SIA doomsday too!
Enter causality
What's going on? SIA was supposed to defeat the doomsday argument! What happens is that I've implicitly cheated - by naming the boxes "2013" and "2014", I've heavily implied that these "boxes" figuratively correspond two subsequent years. But then I've treated them as independent for SIA, like two literal distinct boxes.
In reality, of course, the contents of two years are not independent. Causality connects the two: it's much more likely that there are many observers in 2014, if there were many in 2013. Indeed, most of the observers will exist in both years (and there will be some subtle SSA issues of "observer moments" that we're eliding here - does a future you count as the same observer or a different one). So causality removes the independence assumption, and though there may be some interesting SIA effects on changes in growth rates, we won't see a real SIA doomsday.
Exit causality
But is causality itself a justified assumption? To some extent it is: some people who think they live in 2013/2014 will certainly be in a causal relationship with people who think they live in 2014/2013.
But many will not! What of Boltzmann brains? What of Boltzmann worlds - brief worlds that last less than a year?
What about deluded worlds: worlds where the background data coincidentally (or conspiratorially) imply that we exist on the planet Earth around the Sun, in 2013 - but where we really exist inside a star circling a space-station a few seconds after the seventh toroidal big bang, or something, and will soon wake up to this fact. What about simply deluded people - may we be alien nutjobs, dreaming we're humans? And of great relevance to arguments often presented here, what if we are short-term simulations run by some advanced species?
All these are possible, with non-zero probability (the probability of simulations may be very high indeed, under some assumptions). All of these break the causal link to 2014 or other future events. And hence all of them allow a genuine SIA doomsday argument to flourish: we should expect that seeing 2014 is less likely than is objectively implied, given that we think we are in the year 2013.
Is anthropic decision theory (ADT) subject to the same doomsday argument? In that form, no. In any situation where causality breaks down, your decisions cease to have consequences, and ADT tosses them aside. But more complicated preferences or setups could bring doomsday into ADT as well.