If the universe were to survive 280 billion years, then that would put us within the first 5% of the universe's lifespan. So, if we take an alpha of 5%, we can reject the hypothesis that the universe will last more than 280 billion years.

That sounds like "Copernican" reasoning (assume you are at a random point in time) rather than "anthropic" reasoning (assume you are a random observer from a class of observers). I'm not surprised the Copernican approach gives daft results, because the spatial version (assume you are at a random point in space) also gives daft results: see here in this thread point 2.

Incidentally, there is a valid anthropic version of your argument: the prediction is that the universe will be uninhabitable 280 billion years from now, or at least contain many fewer observers than it does now. However, in that case, it looks like a successful prediction. The recent discovery that the stars are beginning to go out and that 95% of stars that will ever form have formed already is just the sort of thing that would be expected under anthropic reasoning. But it is totally surprising otherwise.

We can also reject the hypothesis that more than 4 trillion humans lives will take place

The correct application of anthropic reasoning only rejects this as a hypothesis about the average number of observers in a civilisation, not about human beings specifically. If we knew somehow (on other grounds) that most civilisations make it to 10 trillion observers, we wouldn't predict any less for human beings.

that any given 1-year-old will reach the age of 20,

That's an instance of the same error: anthropic reasoning does NOT reject the particular hypothesis. We already know that an average human lifespan is greater than 20, so we have no reason to predict less than 20 for a particular child. (The reason is that observing one particular child at age 1 as a random observation from the set of all human observations is no less probable if she lives to 100 than if she lives to 2).

The probability that the right sperm would fertilize the right egg and I would be conceived is much less than 1 in a billion, but that doesn't mean I think I need a new model

Anthropic reasoning is like any Bayesian reasoning: observations only count as evidence between hypotheses if they are more likely on one hypothesis than another. Also, hypotheses must be fairly likely a priori to be worth considering against the evidence. Suppose you somehow got a precise observation of sperm meeting egg to make you, with a genome analysis of the two: that exact DNA readout would be extremely unlikely under the hypothesis of the usual laws of physics, chemistry and biology. But that shouldn't make you suspect an alternative hypothesis (e.g. that you are some weird biological experiment, or a special child of god) because that exact DNA readout is extremely unlikely on those hypotheses as well. So it doesn't count as evidence for these alternatives.

The probability of being born prior to a galactic-wide expansion may be very low, but someone has to be born before the expansion. What's so special about me, that I should reject the possibility that I such a person?

If all hypotheses gave extremely low probability of being born before the expansion, then you are correct. But the issue is that some hypotheses give high probability that an observer finds himself before expansion (the hypotheses where no civilisations expand, and all stay small). So your observations do count as evidence to decide between the hypotheses.