Pater rogat utrum sal in domo iam sit an sal adhuc desit.

Because utrum ... an ... introduces an indirect question after rogat (Father asks whether...). In Latin, the verb in an indirect question is normally subjunctive, so sit = is (subj.) / might be / whether it is rather than the direct-question form est.

utrum ... an ... sets up a two-alternative indirect question: whether A ... or B ...
Here: utrum sal ... sit an ... desit = whether the salt is already in the house or whether it’s still lacking.

By contrast, num usually introduces a yes/no indirect question without explicitly giving the alternative (often with an expectation of no): he asks whether (perhaps not)....

It can be omitted, but repeating it is perfectly normal and often clearer. Latin sometimes repeats the key noun in each alternative to avoid ambiguity:

• full: utrum sal ... sit an sal ... desit
• shorter: utrum sal ... sit an adhuc desit (still understandable)

Repetition also gives a more balanced A / B structure.

sal is nominative singular. It’s the subject of both verbs:

• sal ... sit = the salt is
• sal ... desit = the salt is lacking

rogat does not take sal as its direct object here. Instead, it takes an entire indirect question clause (utrum ... an ...) as what is being asked.

desit is the present subjunctive of desum, deesse, defui (to be absent / be lacking).
Literally, sal desit = the salt is absent or the salt is lacking.

It’s subjunctive for the same reason as sit: it’s inside an indirect question.

in domo uses in + ablative to mean in (location where).
Yes, domi is also possible and is very common: domi = at home / in the house.

So you could also say: Pater rogat utrum sal domi iam sit an sal adhuc desit.
(domi is a locative form.)

They set up contrasting time perspectives:

• iam = already (the expected state has happened)
• adhuc = still / up to now (the expected change has not happened yet)

So iam sit suggests: maybe it has arrived already, while adhuc desit suggests: maybe it’s still missing.

Latin word order is flexible, but it often places:

• key nouns early (sal) to mark the topic,
• adverbs near what they modify (iam near sit, adhuc near desit),
• verbs toward the end of their clause.

You could rearrange without changing the basic meaning, e.g.:

• ... utrum iam sal in domo sit an adhuc desit. The original order is just a natural, clear way to present it.

No. Latin an in utrum ... an ... means or in a two-part question.
It’s unrelated to English and; it’s more like whether ... or ....

Here pater rogat means Father asks in the sense of inquires/wonders, and what he asks is the indirect question that follows. No direct addressee is expressed.

If Latin meant Father asks someone (requests something from someone), you’d typically see an expressed person (often accusative) and/or a different construction, e.g. patrem rogat (he asks the father) or rogat ut... (he asks that...).