Domina sacculum tabellarii aperit et medicamentum invenit.

Yes. domina is nominative singular, so it’s the subject: the lady / mistress. It’s a 1st-declension noun (base form domina, -ae).

Because it’s the direct object of aperit (opens), so it’s in the accusative singular.

• Nominative singular: sacculus (subject form)
• Accusative singular: sacculum (object form)
  It’s a 2nd-declension masculine noun (base form sacculus, -ī).

tabellarii is genitive singular, meaning of the courier / of the letter-carrier. It shows possession: sacculum tabellarii = the courier’s bag.

Yes. Latin often uses the genitive for possession (as here), but it can also use a possessive adjective:

• sacculum eius = his bag (using the pronoun eius)
• sacculum meum = my bag
  But tabellarii is more specific: it tells you whose bag by naming the person.

Both are present tense, 3rd person singular:

• aperit = (she) opens (from aperīre)
• invenit = (she) finds (from invenīre)

A common clue is the -t ending, which often marks he/she/it in the present tense.

Latin often omits repeated subjects when it’s clear they stay the same. Since domina is already the subject of aperit, it’s naturally understood as the subject of invenit as well: (she) opens … and (she) finds ….

et simply means and, coordinating two verbs/actions:

• aperit … et invenit
  So the sentence reports two actions by the same subject.

medicamentum is the direct object of invenit (finds), so it’s accusative singular. It’s a 2nd-declension neuter noun (base form medicamentum, -ī). Neuter accusative singular looks the same as neuter nominative singular: -um.

Latin word order is fairly flexible because the endings show grammatical roles. You could reorder for emphasis, for example:

• Sacculum tabellarii domina aperit et medicamentum invenit.
  Still: The lady opens the courier’s bag and finds medicine.
  That said, the given order is very natural: subject → object → verb.

In (restored) Classical-style pronunciation:

• tabellarii ≈ ta-bel-LA-ree-ee (with a rolled/flipped r in many traditions)

It comes from tabellarius, meaning a letter-carrier/courier (connected with tabella, a small tablet/writing board, hence written messages).