Ianua villae aperta est, et nos intramus.

Because ianua is in the nominative singular and it agrees with the verb phrase aperta est (has been opened / is open). In Latin, the subject is typically shown by case (nominative), not by word order. So ianua = the door is the thing being described as aperta.

Here villae is genitive singular, meaning of the villa: ianua villae = the door of the villa. (The dative singular of villa is also villae, but the meaning “to/for the villa” wouldn’t fit well with door; genitive possession is the normal reading.)

Aperta est is the perfect passive of aperire (to open): literally has been opened, often best translated as is open (a resulting state).

• aperit would mean (someone) opens (active, present).
• aperitur would mean is being opened / gets opened (passive, present).
  So aperta est focuses on the door being in the opened state.

Because aperta is a participle/adjective agreeing with ianua, which is feminine nominative singular. Agreement in Latin is by gender, number, and case:

• ianua (fem. nom. sg.)
• aperta (fem. nom. sg.)

Both are possible depending on context:

• was opened = a past event (perfect passive)
• is open = a present state resulting from a past action
  With a following action et nos intramus (and we enter), it often makes good sense as “The door is open, and we enter.” But grammatically “has been opened” is also accurate.

Et connects two independent clauses: 1) Ianua villae aperta est = The door of the villa has been opened / is open.
2) nos intramus = we enter.
So it’s simply “and” joining two statements.

Yes—intramus already contains we (1st person plural). Nos is added for emphasis or contrast, like “and we enter” or “and we (as opposed to others) enter.” Latin often omits subject pronouns unless they add something.

Intramus is present indicative active, 1st person plural, from intrare (to enter): we enter / we are entering.

Intrare is commonly used intransitively (no direct object) meaning to go in / enter. The destination can be implied, or expressed with something like:

• in villam intramus = we enter into the villa (accusative after in)
  But it’s perfectly normal to say simply nos intramus when the place is obvious from context.

Latin word order is flexible. Ianua villae comes first to set the topic (the door of the villa), and aperta est completes the idea. You could also say Aperta est ianua villae for a slightly different emphasis (more like “Open is the door...”). Case endings keep the grammar clear either way.

Villa often suggests a country house/estate or a larger property, while domus usually means a home/house (often an urban household) and can carry a more “family home” feel. Both can be translated house, but the nuance differs.

Yes. Ianua can mean a door or an entrance, and sometimes a gate (especially as a main entrance). Context decides. If it were clearly a gate, Latin might also use porta, but ianua is still possible.

Because genitives commonly attach to a nearby noun to show possession, and ianua villae is a very natural “noun + genitive” phrase (door of the villa). Also, aperta est doesn’t take a genitive, so villae has nowhere else sensible to belong.

Not necessarily. The agent (by X) can be added in Latin with:

• ab
  • ablative for a person: ab servo aperta est = it was opened by the slave
    But if it’s not included, the sentence simply states the passive fact/state without naming an agent.