Pater dicit multos Romanos ad Tiberim vesperi ambulare.

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

In English, we say:

• Father says that many Romans are walking...

In Latin, after verbs like dicit, putat, scit, and so on, the person or thing that is the subject of the reported statement usually goes into the accusative, and the verb of that reported statement goes into the infinitive.

So:

• multos Romanos = the subject of the reported idea, but in the accusative
• ambulare = to walk, used as the infinitive in indirect statement

This is often called the accusative-and-infinitive construction.

For the same reason: after dicit, Latin often reports what someone says by using:

• accusative for the subject of the reported statement
• infinitive for the verb of the reported statement

So instead of saying something equivalent to:

• Pater dicit quod multi Romani ad Tiberim vesperi ambulant

classical Latin much more naturally says:

• Pater dicit multos Romanos ad Tiberim vesperi ambulare

Literally, this is something like:

• Father says many Romans to walk to the Tiber in the evening

But in good English we translate it as:

• Father says that many Romans walk / are walking to the Tiber in the evening.

Latin usually does not need a separate word meaning that in this kind of sentence.

English often says:

• Father says that many Romans...

Latin usually expresses this idea by structure rather than by a separate word:

• dicit
  • accusative subject
    • infinitive verb

So the idea of English that is built into the construction itself.

The cases and verb forms tell you.

• pater is nominative singular, so it is the subject of the main verb dicit
• dicit = says
• multos Romanos is accusative plural, which shows it is the subject of the infinitive ambulare inside the indirect statement

So the structure is:

• pater → subject of dicit
• multos Romanos → subject of ambulare

In other words:

• Father says
• many Romans are walking

Even though multos Romanos is not nominative, it still functions as the logical subject of the reported action.

Because ad takes the accusative case when it means motion toward a place.

Here the destination is the river Tiber:

• dictionary form: Tiberis
• accusative singular: Tiberim

So:

• ad Tiberim = to the Tiber

This is normal Latin:

• ad
  • accusative = to / toward

If you used Tiberis, that would not fit after ad here.

Vesperi means in the evening or toward evening, depending on context.

It is an old time expression used adverbially. For a learner, the simplest way to understand it is:

• vesperi = in the evening

Latin often uses certain case forms or fossilized expressions to indicate time when something happens.

So in this sentence:

• vesperi ambulare = to walk in the evening

You may also meet related forms such as vespere. For this sentence, it is enough to recognize vesperi as a standard temporal expression.

Latin word order is much more flexible than English word order because Latin uses case endings to show grammatical function.

So this sentence can place words in an order that sounds natural in Latin:

• Pater dicit multos Romanos ad Tiberim vesperi ambulare.

A few things are going on:

• pater comes first as the topic or subject
• dicit appears early, so you quickly know this is a statement about what someone says
• multos Romanos introduces the reported subject
• ad Tiberim and vesperi add place and time
• ambulare comes at the end, which is a very common position for an infinitive in indirect statement

So yes, the order is meaningful stylistically, but the grammar is mainly shown by endings, not position.

Because classical Latin has no articles.

So:

• pater can mean father, the father, or sometimes a father, depending on context
• Romanos can mean Romans or the Romans
• Tiberim can mean the Tiber

English must choose an article, but Latin usually leaves that to context.

Most naturally, it means many Romans.

• multos = many
• Romanos = Romans

If Latin wanted to say many of the Romans, it would usually express that more explicitly, often with a partitive idea.

So here the straightforward meaning is simply:

• many Romans

Yes. The Latin present tense can often be translated in more than one way depending on context.

So dicit could be:

• says
• is saying
• sometimes even does say, if emphasis is needed

In a simple sentence like this, says is usually the best default translation.

Yes. Since ambulare is a present infinitive, it shows action happening at the same time as dicit.

Depending on context, English may render that as:

• Father says that many Romans walk to the Tiber in the evening
• Father says that many Romans are walking to the Tiber in the evening

Both can be reasonable. English chooses between simple and progressive more often than Latin does, so Latin is sometimes less specific here.