Short answers, pulled from the story.
A theorem is a statement that has been proven true using the inference rules of a deductive system, starting from axioms and previously proven theorems. It is distinct from a conjecture, which is an unproven statement believed to be true, and from a scientific theory, which is validated by experiment rather than by pure logical deduction.
A theorem is an important proven result. A lemma is an accessory proposition with little applicability outside a specific proof, though some lemmas, such as Gauss's lemma and Zorn's lemma, have grown in significance over time. A corollary is a proposition that follows immediately from a theorem or axiom with little or no additional proof required.
Godel's incompleteness theorems show that every consistent formal theory containing the natural numbers has true statements about natural numbers that cannot be proven within that theory. A concrete example is Goodstein's theorem, which can be stated in Peano arithmetic but cannot be proven there, though it is provable in Zermelo-Fraenkel set theory.
The classification of finite simple groups is regarded by some as the longest proof of a theorem. It spans tens of thousands of pages across approximately 500 journal articles written by roughly 100 authors. Several ongoing projects aim to shorten and simplify this proof.
The aphorism that a mathematician is a device for turning coffee into theorems is probably due to Alfréd Rényi, though it is often attributed to his colleague Paul Erdos. Erdos was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.
A mathematical theorem is proven by pure logical deduction from axioms; no experiment can validate or overturn it. A scientific theory is falsifiable, meaning it makes testable predictions about the natural world, and any disagreement between prediction and experiment can demonstrate its incorrectness or limits.