Learn Portuguese (Portugal): O miúdo está contente com o presente.

We use está (from the verb estar) because being pleased is a temporary feeling or state right now. If we used é (from the verb ser), it would mean the kid has a permanently happy personality.

I usually see criança or menino for child. Is miúdo commonly used?

Yes, miúdo is extremely common in European Portuguese for a kid or little boy in everyday, informal speech. If it were a little girl, you would say miúda.

If the sentence was about a girl, would contente change to contenta?

No, it stays exactly the same. Adjectives that end in -e, like contente or triste (sad), are used for both masculine and feminine nouns. So it would simply be A miúda está contente.

Does presente only mean a gift?

Just like in English, presente can mean a gift or the present time. It also translates to present in the sense of being physically at an event, like answering a school roll call. Here, the context with the word com (with) makes it clear it is a gift.