Avus dicit nos in praesenti vivere debere.

Because this sentence uses indirect statement after dicit (says).

In Latin, after verbs of saying, thinking, knowing, and perceiving, the person who would be the subject of the reported statement usually goes into the accusative, and the verb goes into the infinitive.

So instead of:

• Nōs vivimus = We live

Latin says:

• dicit nōs vivere = he says that we live

So nos is not the subject of the main verb dicit. It is the accusative subject of the infinitives vivere and debere.

It is an accusative-and-infinitive construction, often abbreviated ACI.

The pattern is:

• a main verb of saying/thinking/etc.
• an accusative noun or pronoun
• one or more infinitives

Here:

• Avus dicit = Grandfather says
• nos = the subject of the reported statement, in the accusative
• vivere debere = the verbal part of the reported statement

So the whole thing means something like:

• Grandfather says [that] we ought to live in the present.

Because they belong to the indirect statement after dicit.

In English, we usually use that plus a finite verb:

• He says that we should live...
• He says that we ought to live...

Latin usually does not do that here. Instead, it uses infinitives:

• nos vivere debere

Also, debere is functioning like a modal idea, so vivere depends on debere:

• debere = to owe, to ought, to must
• vivere debere = to ought to live / to be supposed to live

So the reported statement is built with infinitives rather than a finite clause.

Debere literally means to owe, but it often develops the sense to ought to, to be supposed to, or should.

So:

• vivere = to live
• debere = to ought / to be obliged
• vivere debere = to ought to live

In smoother English, we would usually translate this as:

• we should live
• we ought to live

So debere adds the idea of necessity, duty, or appropriateness.

Because that would be a direct statement, not an indirect one.

• Nos debemus vivere = We ought to live
• Avus dicit nos debere vivere = Grandfather says that we ought to live

After dicit, Latin normally changes the reported clause into the accusative-and-infinitive pattern. That is why debemus becomes debere, and nos becomes accusative.

In praesenti means in the present, or more naturally, in the present moment / in the here and now.

Breaking it down:

• in = in
• praesenti = ablative form meaning the present

Because in with location takes the ablative, praesenti is in the ablative.

So in praesenti vivere means:

• to live in the present
• to live in the present moment

Because it follows the preposition in in a sense of location or state.

Latin uses:

• in
  • ablative for in / on something
• in
  • accusative for into / onto something

Here the meaning is not movement into the present, but being or living in it. So Latin uses the ablative:

• in praesenti = in the present

The subject of dicit is Avus.

• Avus = grandfather
• dicit = says

So:

• Avus dicit = Grandfather says

Then nos is not the subject of dicit; it belongs to the indirect statement.

Yes, the word order is normal enough, and Latin word order is much more flexible than English word order.

This sentence is arranged as:

• Avus = subject
• dicit = main verb
• nos ... vivere debere = indirect statement
• in praesenti placed near vivere, where it makes good sense

Latin could also say things like:

• Avus nos in praesenti vivere debere dicit.
• Nos in praesenti vivere debere avus dicit.

The meaning would stay basically the same, though emphasis might shift. The given order is clear and natural.

Because the main statement is Grandfather says.

• dicit = he says, present tense
• dixit would mean he said
• dicet would mean he will say

So the sentence presents the act of saying as present-time.

Also, in an indirect statement, the tense of the infinitive does not work exactly like an English tense. The present infinitive usually shows action that is simultaneous with the main verb. So:

• dicit nos vivere debere suggests he says that we ought to live / he says that we should live

Avus is nominative singular because it is the subject of the main verb dicit.

Basic structure:

• Avus = nominative subject
• dicit = main verb
• nos = accusative subject of the infinitive construction

A learner might expect both avus and nos to be subjects, but they belong to different levels of the sentence:

• avus is the subject of the main clause
• nos is the subject of the reported clause

Yes. In fact, should is often the most natural English translation here.

Latin debere can be translated in several ways depending on context:

• ought to
• should
• must (sometimes, depending on strength)
• be supposed to

So:

• nos in praesenti vivere debere could be rendered as
  • that we ought to live in the present
  • that we should live in the present

Even if the Latin word is not literally should, that is often the best English equivalent.

It means to ought to live or to have to live, not to be living.

That is because:

• vivere = to live
• debere adds obligation or necessity

So this is not a progressive idea like to be living. It is a modal idea:

• to need to live
• to ought to live
• to be obliged to live

In context, ought to live or should live is probably the best choice.

Because in this construction Latin normally does not need one.

English often says:

• Grandfather says that we ought to live in the present.

Latin instead uses the accusative-and-infinitive construction, which replaces the whole that-clause:

• Avus dicit nos in praesenti vivere debere.

So the idea of that is built into the grammar of the sentence rather than expressed by a separate word.