Learn Italian: Il gatto dorme sul balcone. - The cat sleeps on the balcony.

sul is a combination of the preposition su (on) and the masculine singular article il (the). Because balcone (balcony) uses il, they merge together into sul.

'Balcone' ends in '-e', so how do we know it needs the masculine article 'il' (in 'sul')?

Nouns ending in -e in Italian can be either masculine or feminine. Balcone happens to be masculine. For words ending in -e, you just have to memorize their gender as you learn them!

Why does 'dorme' end in '-e' if it's the 'he/she/it' form?

Dorme comes from the verb dormire (to sleep). It is a regular -ire verb, and the 'he/she/it' ending for -ire verbs (and -ere verbs) is -e.

Can this sentence also mean 'The cat is sleeping on the balcony'?

Yes! The Italian present tense covers both the simple present ('sleeps') and the present continuous ('is sleeping'). You will usually understand which one is meant from the context.