You are skeptical of the claim that "everything will occur, given an infinite number of opportunities." Other answers have given a good explanation of when this claim is true and when it is false. However, I would like to assemble the various ideas into a single answer.
Probability problems are often formulated in terms of choosing marbles from an opaque jar, which is valuable because it appeals to our intuition, to the extent that it can. The marbles represent the space of all possible outcomes (or: all possible values for the random variable). Picking a marble corresponds to sampling the space.
Now, there are two ways to conduct a sample: with replacement, and without replacement. After you pull out a marble, do you keep it, or do you put it back before pulling out another marble? The Gambler's Fallacy is nothing more than the mistaken idea that all probabilities (or, at least the ones of interest) entail sampling without replacement. Or, to illustrate more clearly, that all games of chance are equivalent to counting down a finite blackjack deck. If roulette involved taking each number off the wheel as it occurs, then the Gambler's Fallacy would actually be true for roulette. And if the dealer always replaced played cards into the shoe (randomly!) after every hand, it would be impossible to usefully count down a blackjack deck (it would become a circular, or "infinite" shoe, although an 8-deck shoe with a deep cut makes for a useful approximation).
When it comes to monkeys on typewriters, we have an additional complication: time. We can view the probabilistic event as a monkey striking a key, or as a monkey producing an entire sequence of keystrokes. In fact, the latter is a far more useful way to view the situation. So instead of putting a marble for each letter of the alphabet into our bag, and trying to keep track of what texts are produced by pulling out thousands of marbles, we can instead inscribe the texts which are produced by all the monkeys after 1 keystroke, after 2 keystrokes, etc. up to the limit of what monkeys are willing or able to type. So one marble will have the text "q" on it, while another will have the text "mxlplx", and yet another will have: "To be or not to be".
Since we are trying to avoid the Gambler's Fallacy, we must sample the bag with replacement. After all, there's nothing stopping a monkey from typing "MonkeyButt" 23 times in a row. So we must be able to draw this marble from the bag at least 23 times, and we can only do that if we put it back. Now, the original question becomes: "Given an unlimited number of draws, are we guaranteed that we will draw a marble with the entire text of Hamlet carefully inscribed upon its surface?" And the answer is: "It depends."
You see, we made a subtle but important leap when we switched the random variable from keys typed to texts typed. We sort of hand-waved away how long the texts could be. In fact, even if we have an infinite number of monkeys, nobody has suggested that the monkeys themselves are immortal, or have infinite patience. It could turn out that no monkey is willing to type more than 10,000 keystrokes, under any circumstances. If that is the case, then we have no chance of drawing Hamlet, no matter how lucky those keystrokes are (unless you are willing to assemble works from multiple monkeys, but that ruins the claim in other ways).
The Outer Limits
All of this is a fancy way to point out what is hopefully by now an obvious fact: you can only draw a marble from the bag, if the marble is already in the bag. If we have theoretically tireless monkeys which are highly motivated to type and physically capable of typing at least as many characters as can be found in Shakespeare, and there are no constraints on the sequences of characters typed (perhaps monkeys don't like to type 'p' after 'a' because they are on the opposite sides of a QWERTY keyboard), then, given an infinite number of "monkey texts", the probability that one of them corresponds to Hamlet is 1.
Now, let's talk about planets. If the forces which affect planet formation have a finite range, and the universe has infinite size, and the universe has infinite matter, and the universe has mostly uniform density (consistent with the observable universe, at least), and the laws of physics are the same everywhere in the universe, then we basically have the physical conditions necessary to create any kind of planet which can be formed under conditions similar to earth. Under those conditions, I would tend to agree that the probability of another earth-like planet existing is 1.
In fact, I would agree that the probability of TEN other earth-like planets is 1. I would go so far as to claim that there are an infinite number of earth-like planets in such a universe. This is due to the simple fact that we as humans can only distinguish a finite number of planets as "different", due to the limitations of physics. Therefore, we can put every "possible-planet marble" into our bag, but our bag will only contain a finite number of marbles, including our "pale blue dot". And since we will draw from the bag an infinite number of times with replacement, it follows that earth and every other kind of planet we have or will observe must occur an infinite number of times.
However, there are several things that we won't see: we won't see a cube-shaped planet, or a donut-shaped planet, or a planet that looks like a Sierpinski triangle. That's because physics does not allow the construction of such planet shapes. So an infinite number of draws does not allow anything at all to happen. It only allows any event which is individually possible to happen, possibly an infinite number of times. You can only draw a marble from the bag if the marble can exist and you put it in the bag.