Learn Farsi: تولدت مبارک!

Yes! Tavallod means "birth" or "birthday", and the pronominal enclitic -at means "your". So tavallodat literally means "your birthday". Just like you are learning to attach these enclitics to prepositions, they are heavily used on nouns to show possession.

Why is there no verb like "is" (ast or e) in this sentence?

In common greetings and well wishes, the verb "to be" is often dropped to keep the phrase short and natural. You could technically say tavallodat mobârak ast (literally "your birthday is blessed"), but native speakers almost always leave the verb out.

What does mobârak literally mean, and can I use it for other occasions?

Mobârak literally means "blessed" or "auspicious", and it is the go-to word for congratulations in Farsi. You will see it in phrases like sâl-e now mobârak (Happy New Year), or you can just say mobârak! by itself to congratulate someone on a new house or a new job.

Is there an ezafe linking the noun and the adjective here?

No, there is no ezafe here because this is not a single descriptive noun phrase like "the happy birthday". It acts as a complete sentence: tavallodat is the subject ("your birthday"), and mobârak is the adjective describing it ("[is] happy").