Learn Russian: Ваш номер уже забронирован и оплачен. - Your room is already booked and paid for.

They are short-form past passive participles. They describe a state resulting from a completed action, acting a lot like short adjectives. They are formed from the perfective verbs забронировать (to book) and оплатить (to pay) to show that the room has had these actions done to it.

Why does забронирован end in -ан but оплачен ends in -ен?

The suffix depends on the infinitive of the original verb. Verbs ending in -ать (like забронировать) form their short participles with -ан (or -ана, -ано, -аны). Verbs ending in -ить (like оплатить) form them with -ен. Because номер is a masculine noun, both are in the masculine singular form.

The translation says "is already booked", but there is no verb for "is" in Russian here. Why?

Just like with regular adjectives and nouns, Russian omits the "to be" verb in the present tense. If this were in the past, you would add был (номер был забронирован = the room was booked), and for the future, you would use будет.