Learn Italian: Queste lettere sono vecchie?

Queste is the feminine plural form of "these". Because the noun lettere (letters) is feminine plural, the word for "these" must match it. If the noun were masculine plural, like ragazzi (boys), you would use questi.

Why does the word vecchie have an "h" in it?

The singular feminine word for "old" is vecchia. To make it plural, it changes to vecchie. The h is added to keep the "c" sounding hard (like a "k"). Without the h, a "c" followed by an "i" or "e" would make a soft "ch" sound in Italian.

Where is the word for "are" at the beginning of the question?

In English, we swap the order to make a question ("These letters are" becomes "Are these letters"). Italian doesn't do this! You just keep the normal statement order (Queste lettere sono...) and simply raise your voice at the end when speaking to show it's a question.

Does lettere mean alphabet letters or letters you write to someone?

Just like in English, it can mean both! A lettera can be a letter in a word or a letter you send in the mail. Here, it likely means the mail, since you are asking if they are old.