Postquam negotium finitum erit, mater supra mensam novem pira et decem nuces ponet, ut omnes laeti cenent.

Postquam means after. It introduces a time clause telling you when the main action will happen.

So the structure is:

• Postquam negotium finitum erit = after the task/business has been finished
• mater ... ponet = mother will place ...

Latin often puts this kind of time clause first, but it does not have to.

Finitum erit is future perfect passive: will have been finished.

Latin often uses the future perfect in a subordinate clause when the main clause is in the future and the action in the subordinate clause must happen first.

So:

• negotium finitum erit = the task will have been finished
• mater ... ponet = mother will place ...

In other words: first the task gets finished, then the mother places the fruit and nuts.

It is made of two parts:

• finitum = perfect passive participle of finire
• erit = future of esse

Together they form the future perfect passive.

Also, finitum agrees with negotium:

• negotium is neuter singular
• finitum is neuter singular

So literally it is something like the task, having been finished, will be → more naturally, the task will have been finished.

Mater is the subject of ponet, so it is in the nominative case.

• mater = mother as the doer of the action
• ponet = will place

If mother were the direct object, you would expect a different case. But here she is the one doing the placing.

Supra is a preposition that takes the accusative here, so mensam is accusative singular.

• supra = above / over / on
• mensam = table in the accusative

So supra mensam means on the table or above the table, depending on context. In this sentence, on the table is the natural English translation.

They are the direct objects of ponet, so they are in the accusative plural.

• pira = pears (accusative plural)
• nuces = nuts (accusative plural)

They are the things the mother will place on the table.

The numbers do not change form here:

• novem = nine
• decem = ten

These numerals are indeclinable in this use.

Here ut introduces a purpose clause.

So:

• ut omnes laeti cenent = so that all may dine happily / so that everyone may eat dinner happy

This tells you the purpose of the mother’s action. She puts the pears and nuts on the table in order that everyone may have a happy meal.

Because after ut in a purpose clause, Latin normally uses the subjunctive.

So:

• ut ... cenent = so that ... may dine

The tense is present subjunctive because the main verb is in a primary tense (ponet, future), and Latin regularly uses the present subjunctive in purpose clauses after a primary tense.

This is not saying they are actually dining yet; it expresses the intended result or purpose.

Omnes is the subject of cenent, and laeti agrees with it.

• omnes = all (people), everyone
• laeti = happy, masculine plural nominative

So laeti is a predicate adjective describing the people who will dine.

Latin often uses masculine plural for a mixed group or for people in general when no specific noun is stated.

Because laeti is an adjective, while laete would be an adverb.

• laeti cenent = they may dine happy / as happy people
• laete cenent would mean they may dine happily

In this sentence, Latin chooses the adjective, describing the state of omnes rather than just the manner of dining.

Here it is nominative plural, because it is the subject of cenent.

You are right that omnes can also be accusative plural in form. The way to tell is by its job in the sentence:

• cenent needs a subject
• omnes is the group doing the dining

So here it must be nominative.

Latin word order is much freer than English word order. A verb often comes near the end of its clause, especially in straightforward prose.

So this order is very natural:

• mater supra mensam novem pira et decem nuces ponet

Even though English prefers Mother will place..., Latin can comfortably save the verb until later.

The same flexibility appears in the whole sentence:

• time clause first
• then main clause
• then purpose clause

Yes, depending on the meaning.

With a past main verb, Latin often uses a perfect or historical tense after postquam. But here the main clause is future:

• ponet = will place

Since the finishing must be completed before that future action, Latin uses:

• finitum erit = will have been finished

So this tense choice is especially appropriate because the sentence is talking about a future sequence: after X has happened, Y will happen.