Learn German: Je länger der Wartende am Schalter steht, desto wütender wird er. - The longer the person waiting stands at the counter, the angrier he gets.

This is a nominalized present participle, which is a verb turned into a noun to describe a person doing an action. It comes from the verb warten (to wait), which becomes the participle wartend (waiting). Because it acts like an adjective that has been turned into a noun, it takes standard adjective endings. Here it is the masculine singular subject following the definite article der, so it gets the weak ending -e.

If a woman were waiting, would the word change?

Yes! Because these nouns follow adjective ending rules, the gender changes the article. The weak ending for a feminine nominative noun is also -e, so a woman waiting would be die Wartende. If you were talking about multiple people waiting (plural), it would be die Wartenden.

Why are the verbs in different places in the two halves of the sentence (steht at the end, but wird in the middle)?

This sentence uses the two-part connector je ... desto ... (the ... the ...). The je half acts as a subordinate clause, which pushes the conjugated verb (steht) to the very end. The desto half acts as a main clause. The phrase desto wütender takes up the first position, so the verb (wird) must take position 2, immediately followed by the subject (er).

Why is it am Schalter (dative) and not (accusative)?