Learn Turkish: Barış hoşgörü ile başlar. - Peace begins with tolerance.

It comes from the compound verb hoş görmek, which literally means "to see pleasantly" but is used to mean "to tolerate." The suffix -ü is added to the verb root gör- to turn it into the noun hoşgörü (tolerance).

Does the verb başlamak (to begin) come from a noun?

Yes! It comes from the noun baş, meaning "head." The suffix -la is a very common denominal suffix that turns a noun or adjective into a verb. So, to "head" something evolved into meaning to "start" it. While your current lesson focuses on the -(I)msEmEk suffix for attitude verbs, -la is another fundamental way Turkish builds verbs from nouns.

Could we attach ile to the preceding word and say hoşgörüyle instead?

Absolutely. Barış hoşgörüyle başlar is perfectly natural and means the exact same thing. Writing ile as a separate word is slightly more formal and emphatic, which fits the proverb-like, philosophical tone of this sentence.

Does the word barış contain the reciprocal suffix -ış?