Learn Italian: Non dormo bene perché ho una brutta tosse. - I don't sleep well because I have a bad cough.

No, in Italian, perché is generally only used in the middle of a sentence to introduce a reason. If you want to put the reason first ("Because I have a bad cough, I don't sleep well"), you must use siccome instead: Siccome ho una brutta tosse, non dormo bene.

Why is brutta placed before tosse instead of after?

Common descriptive adjectives like bello, brutto, buono, and cattivo usually come before the noun they describe. Placing brutta before tosse sounds much more natural here than saying una tosse brutta.

How do I know the gender of tosse since it ends in -e?

Words ending in -e can be either masculine or feminine, so you just have to memorize their gender when you learn them. Tosse is feminine, which is why it uses the feminine article una and the feminine adjective brutta.

Why isn't there an io before dormo or ho?