Formularul arată vârsta.

Because -ul is the Romanian definite article for this masculine/neuter singular noun.
So:

• formular = form / a form
• formularul = the form

Unlike English, Romanian usually attaches the to the end of the noun instead of putting it before the noun as a separate word.

For the same basic reason: it is the definite form, meaning the age.

• vârstă = age / an age
• vârsta = the age

This noun is feminine, and many feminine nouns form the singular definite article with -a.

arată is the 3rd person singular present tense of the verb a arăta.

Here it means shows or displays.

So grammatically it matches formularul:

• formularul = the form
• arată = shows

Romanian uses the same 3rd person singular verb form for he, she, and it, so arată can mean he shows, she shows, or it shows, depending on context.

The dictionary form is a arăta.

That is the infinitive, equivalent to English to show or sometimes to look / to appear, depending on context.

In this sentence, the meaning is clearly to show/display.

Yes. a arăta is a common verb with a few related meanings, such as:

• to show
• to point out
• to look / appear in some contexts

For example:

• El arată bine. = He looks good.
• Graficul arată rezultatele. = The chart shows the results.

In Formularul arată vârsta, the natural meaning is The form shows the age.

Because Romanian usually uses a postposed definite article: it is attached to the noun.

So instead of:

• English: the form, the age

Romanian has:

• formularul
• vârsta

This is one of the big structural differences between English and Romanian.

Yes. This is a normal subject–verb–object order:

• Formularul = subject
• arată = verb
• vârsta = object

So it lines up nicely with English:

• The form / shows / the age

Romanian word order can be more flexible than English, but this sentence uses the standard, neutral order.

A rough English-friendly guide would be:

for-moo-LAH-rul uh-RAH-tuh VEER-sta

A few notes:

• ă is a short central vowel, something like the a in sofa
• â is a special Romanian vowel with no exact English match; it is a central sound somewhat deeper than ă
• stress is roughly:
  • formuLArul
  • aRAtă
  • VÂRsta

If you want to sound more natural, pay special attention to ă and â, because they are important in Romanian pronunciation.

They are the same noun in different forms:

• vârstă = indefinite form, age / an age
• vârsta = definite form, the age

This is similar to the difference between carte and cartea, casă and casa, etc.

Normally, no—not in this meaning.

Formularul arată vârsta is the natural way to say The form shows the age.

Using bare vârstă here would sound incomplete or unnatural in standard Romanian, because you are referring to a specific piece of information: the age.

It is neuter.

In Romanian, neuter nouns behave:

• like masculine in the singular
• like feminine in the plural

So in the singular you get:

• un formular
• formularul

That is why it looks masculine in this sentence.

Because Romanian often leaves subject pronouns out when they are not needed.

Here, the noun formularul is already present, so there is no reason to add a separate pronoun.

English says:

• The form shows the age

Romanian also simply says:

• Formularul arată vârsta

There is no need for an extra word meaning it.