Avus dicit puerum cautum esse debere, cum per scalas descendat.

Because after dicit Latin uses an indirect statement construction: accusative + infinitive.

So instead of saying:

• Grandfather says that the boy ought to be careful

Latin says, more literally:

• Grandfather says the boy to ought to be careful

That is why puerum is accusative: it is the subject of the infinitive phrase cautum esse debere.

Because cautum agrees with puerum.

The adjective is cautus, cauta, cautum = careful, cautious.
Since puerum is:

• masculine
• singular
• accusative

the adjective must also be:

• masculine
• singular
• accusative

So:

• puer cautus = the careful boy / the boy is careful
• puerum cautum = the boy as accusative, with careful agreeing with it

In this sentence, cautum is a predicate adjective inside indirect statement.

Because each infinitive has a different job.

• debere = to ought / to have to
• esse = to be

And debere normally takes another infinitive with it. So:

• esse debere = to ought to be

Inside indirect statement, the finite verb debet becomes infinitive debere.
Then esse stays as the infinitive that goes with debere.

So puerum cautum esse debere means:

• literally: the boy to ought to be careful
• naturally: that the boy ought to be careful

Yes. Latin word order is flexible.

All of these are basically the same in sense:

• puerum cautum esse debere
• puerum debere esse cautum

The given order puts cautum earlier and leaves debere at the end of the indirect statement, which is very normal in Latin.

A good way to unpack the sentence is:

1. Avus dicit = Grandfather says
2. puerum cautum esse debere = that the boy ought to be careful
3. cum per scalas descendat = when/while he goes down the stairs

No. Here cum is a conjunction, not the preposition with.

It means when or while:

• cum per scalas descendat = when/while he goes down the stairs

So this is not:

• with the stairs
• with descending

It introduces a time/circumstance clause.

There are two helpful ways to think about it:

1. Cum clauses describing the circumstances of an action often use the subjunctive.
2. This clause is also part of the reported content after dicit, and subordinate clauses in reported speech commonly appear in the subjunctive.

So descendat is not surprising here. It gives the sense:

• when/while he is going down the stairs
• or more generally whenever he goes down the stairs

The important point for a learner is: after cum, especially in this kind of clause, Latin often uses the subjunctive rather than the indicative.

Per takes the accusative, so scalas is accusative plural.

Literally, per often means through, along, or by way of.
With scalas and a verb like descendere, the phrase means naturally:

• down the stairs

So per scalas descendat is the normal idea of descending by means of the stairs, not something awkward like through the stairs.

Also, Latin often uses scalae/scalas in the plural, just as English commonly says stairs.

The subject is the boy.

Latin often leaves the subject unstated when it is clear from context. Here the person who ought to be careful is also the person going down the stairs.

So the understood subject of descendat is the same boy referred to by puerum in the indirect statement.

Historically, cautus comes from the verb cavere and is a past participle in origin. But in ordinary Latin it is very commonly used simply as an adjective meaning:

• careful
• cautious

In this sentence, it is best understood as an adjective describing the boy’s state:

• the boy ought to be careful

You do not need to translate it as a literal participle such as guarded here.

The core statement would be:

• Puer cautus esse debet. = The boy ought to be careful.

When that becomes indirect after dicit:

• puer becomes puerum
• debet becomes debere

So:

• Puer cautus esse debet
  becomes
• Avus dicit puerum cautum esse debere

That is a very useful pattern to recognize in Latin.