If you flip a coin an infinite amount of times, where heads and tails are equal probability, then there exists some sequence of flips where all tails will show up (e.g. T,T,T,T...) . It's infinitesimally small, but it exists. However this logic seems backwards to me. If we flip a coin an infinite amount of times it seems there should exist no scenario where a heads simply can not come up. There exists a sequence ${T,T,T,T}$... but this sequence simply doesn't exist within any possible outcomes of the coin flip game.
Another example given on wikipedia is the dart game, where if you have a unit square with each point having an equal probability of being hit, the odds of any single point being hit is 0, however it's still technically possible that you can hit that point. However with our coin flipping game, the sequence is infinite. I understand the probability space of the dart game is infinite, however by definition we have to pull one point so it can not be impossible for any point to be hit. But with the coin game is seems impossible for any one ending sequence to occur, simply because there will never be an ending sequence to our coin flips. Since there will never be any ending sequence it seems impossible to say that there exists a sequence without a heads in it.
Yes. You can leave out from our sample space the sequence $(T,T,T,\dots)$, and that would not be a problem.
In fact, more is true. Let $(x_1,x_2,x_3,\dots)$ be any fixed sequence of $H$ and $T$. Then getting exactly that sequence has probability zero. We can leave that one out of our sample space, and it would be no problem. Any one sequence we can leave out. But, of course, we cannot leave all of these out simultaneously, because there would be nothing left.