Learn French: Ils écrivent un premier brouillon.

Écrire (to write) is an irregular verb. While the singular forms (j'écris, tu écris, il écrit) don't have a 'v', all the plural forms add one: nous écrivons, vous écrivez, ils/elles écrivent.

Do I pronounce the -ent at the end of écrivent?

No, the -ent ending for the 'they' form (third-person plural) of French verbs is always completely silent. You just pronounce the word up to the 'v' sound: 'ay-kreev'.

Why is premier placed before the noun brouillon instead of after?

While most French adjectives go after the noun, adjectives that describe order, numbers, or ranking (like premier for 'first' and dernier for 'last') always go before the noun.

Is there a trick to remembering the gender of brouillon?

Yes! Nouns ending in -illon (like un papillon for 'butterfly') are almost always masculine. You can also tell it is masculine here because it uses un and the masculine adjective premier (rather than the feminine première).