Learn French: Je respire profondément pendant une demi-heure.

This is the special rule for demi. When placed before a noun, demi is invariable (it never changes to match the noun's gender or number) and is always attached with a hyphen. However, when it comes after the noun, it agrees in gender (but never number). If you breathed for an hour and a half, you would say: une heure et demie.

The adjective for "deep" is profond. Why does the adverb profondément have an accent on the é?

Normally, you form an adverb by adding -ment to the feminine adjective (profonde), which would give profondement. However, profond is an exception. It is one of several adjectives where the silent e changes to an é (with an acute accent) in the adverb form. Other common examples are énorme becoming énormément, and précis becoming précisément.

The translation says "for a half-hour". Why use pendant instead of pour here?

English speakers often overuse pour because it translates directly to "for". In French, pendant is used to express the actual duration of an action (similar to "during"). You only use pour for a duration of time when talking about a projected or intended duration in the future, such as: Je pars pour une semaine (I am leaving for a week).