Dominus servum iubet ianuam claudere, ut domus tuta sit.

In Latin, verbs like iubeō (iubet = “orders”) commonly use an accusative + infinitive construction:

• Dominus = the subject (the one giving the order)
• servum = the person being ordered (accusative “object” of the ordering)
  So servum is not the grammatical subject of iubet; it’s the person commanded to do something.

It’s the standard accusative + infinitive after a verb of commanding:

• iubet = “he orders”
• servum (acc.) = “the slave” (the one ordered)
• ianuam claudere (infinitive phrase) = “to close the door”
  So literally: “The master orders the slave to close the door.”

Because after iubeō Latin typically expresses what is commanded with an infinitive.
You’re not getting “the slave closes the door” (claudit), but “the master orders (him) to close the door” (claudere).

Because it’s the direct object of claudere (“to close”). The door is what gets closed, so ianua becomes ianuam.

Latin often doesn’t use a separate pronoun when the noun is already there.
servum already tells you who is being ordered, so there’s no need for eum (“him”).

Latin word order is flexible because endings show grammatical roles.
Dominus servum iubet… and Dominus iubet servum… mean essentially the same thing. The given order can put a bit of emphasis on servum (“the slave” as the commanded person), but it’s not a different grammar.

ut introduces a purpose clause (“in order that / so that”).
The sentence has:

• command: Dominus servum iubet ianuam claudere
• purpose: ut domus tuta sit (“so that the house may be safe”)

Because ut purpose clauses use the subjunctive in Latin.
So ut … sit = “so that … may be.”
If you used est, it would sound like a plain statement (“so that the house is safe”), not a purpose.

In purpose clauses, the present subjunctive usually shows an action/state that is intended and generally contemporaneous or subsequent to the main verb.
Here the idea is: the door is to be closed so that the house will be / may be safe.

Because domus is the subject of sit, so it stays nominative: domus.
tuta is a predicate adjective describing domus, so it also stays nominative and agrees with domus in gender/number/case:

• domus (fem. sg. nom.)
• tuta (fem. sg. nom.)

Here domus is nominative singular (“the house”) as the subject of sit.
domus is somewhat irregular in declension (it mixes forms), but you don’t see the irregularity in this particular form; domus is a normal-looking nominative singular.

In meaning, it expresses the purpose of the commanded action (closing the door), but grammatically it’s attached to the whole idea of the command:
“He orders the slave to close the door so that the house may be safe.”
So it explains why the order/action is given.

It’s common (especially in teaching materials) to separate the main clause from the ut purpose clause with a comma for clarity. Latin manuscripts didn’t originally use punctuation like modern English, but in edited/modern Latin a comma here is normal and helpful.