Scriba testamentum lente legit et quaerit quis heres futurus sit.

Because it is the direct object of legit. The scribe is reading the will, so testamentum is the thing being read.

• scriba = subject, nominative singular
• testamentum = direct object, accusative singular
• legit = reads

So the basic structure is: The scribe reads the will.

Because lente is an adverb, and it modifies the verb legit.

• lentus, -a, -um = slow as an adjective
• lente = slowly as an adverb

Here Latin needs the adverb, because the sentence is describing how the scribe reads.

So:

• scriba lentus = a slow scribe
• scriba lente legit = the scribe reads slowly

Formally, legit can be ambiguous:

• present: he/she reads
• perfect: he/she read or has read

In this sentence, the context shows it is present:

• Scriba testamentum lente legit et quaerit...
• The scribe reads the will slowly and asks/wonders...

Since quaerit is clearly present, legit is naturally understood as present too.

Because Latin often does not repeat the subject when it stays the same.

The subject of both verbs is scriba:

• scriba ... legit
• (scriba) quaerit

English often does the same:

• The scribe reads the will slowly and asks...

We do not need to repeat the scribe before asks.

Because this is an indirect question.

The direct question would be:

• Quis heres erit? = Who will be the heir?

When that question is reported after a verb like quaerit (asks, inquires, wonders), Latin turns it into an indirect question:

• quaerit quis heres futurus sit

So the sentence means something like:

• He asks who will be the heir
• He wonders who is going to be the heir

Because quis here is the interrogative pronoun meaning who?

In indirect questions, Latin uses the same interrogative word that would appear in the direct question:

• Quis heres erit? = Who will be the heir?
• quaerit quis heres futurus sit = he asks who will be the heir

By contrast, qui is often a relative pronoun meaning who/which/that:

• vir qui venit = the man who comes

So here Latin needs quis, not qui, because this is a question, not a relative clause.

Because indirect questions in Latin normally take the subjunctive.

So after quaerit, the clause quis heres futurus sit uses sit, not erit.

Compare:

• Direct question: Quis heres erit?
• Indirect question: quaerit quis heres futurus sit

This is one of the most important Latin rules to learn:
reported questions usually use the subjunctive.

It is a way of expressing future time inside the indirect question.

• futurus, -a, -um is the future active participle of sum
• sit is the subjunctive of sum

Together, futurus sit means roughly:

• will be
• is going to be

Since Latin cannot simply use erit in this indirect question, it uses futurus sit instead.

So:

• quis heres erit? = direct question, who will be the heir?
• quis heres futurus sit = indirect question, who will be the heir

Because it is a predicate nominative with quis and sit.

In a sentence with to be, both the subject word and the noun identifying it are usually nominative:

• quis = who
• heres = heir
• sit = may be / is in the indirect-question construction

So quis heres futurus sit literally means:

• who may be going to be heir

More naturally:

• who will be the heir

Since heres describes or identifies quis, it stays in the nominative.

Latin word order is more flexible than English word order because the endings show the grammar.

This order is natural and clear:

• quis = introduces the question
• heres = the predicate noun
• futurus = agrees with quis and points forward to future time
• sit = verb at the end, which is very common in Latin

So the clause builds toward the verb in a very typical Latin way. English has to be more rigid:

• who will be the heir

Latin can move words around more freely without changing the basic meaning.

All of those are possible depending on context.

The basic meaning of quaero is seek, look for, ask, or inquire. With an indirect question like quis heres futurus sit, it usually means something like:

• asks
• inquires
• tries to find out
• wonders

So here quaerit quis heres futurus sit can be understood as:

• he asks who will be the heir
• he tries to find out who will be the heir
• he wonders who will be the heir

The grammar stays the same whichever English wording you choose.