Echa un vistazo a la lista.

Not literally. Echar often does mean to throw, to pour, or to put, but in echar un vistazo it forms an idiomatic expression meaning to take a look or to have a quick look.

So in this sentence, you should learn echar un vistazo as a whole chunk.

Because echa here is an affirmative tú command.

• echas = you throw / you take in the normal present tense
• echa = take! / have! when telling one person informally to do it

So:

• Tú echas un vistazo = You take a look
• Echa un vistazo = Take a look

Yes. Echa is the command form used with tú, so it is informal singular.

In Spain, the other common versions would be:

• Eche un vistazo a la lista. = formal singular
• Echad un vistazo a la lista. = informal plural (vosotros)
• Echen un vistazo a la lista. = formal plural, or plural in Latin American usage

Un vistazo means a glance or a quick look.

That is why echar un vistazo sounds a bit like:

• take a quick look
• have a look
• give it a glance

It usually suggests a look that is brief or casual, not necessarily a long, detailed examination.

Because the expression is echar un vistazo a algo — literally, to cast a glance at something.

So the a is required by the structure of the expression. It is not the personal a used with people; it is just the preposition that goes with this phrase.

Examples:

• Echa un vistazo al informe.
• Echa un vistazo a la página web.
• Echa un vistazo a la lista.

Because Spanish often uses the definite article where English might not.

Here, la lista means the list, usually referring to a specific list that is already known from context.

Spanish generally prefers the article with nouns in cases like this, so Echa un vistazo a la lista sounds natural, while dropping the article would usually sound incomplete or unnatural unless there were a special context.

Yes, you could, but the nuance is a little different.

• Mira la lista = Look at the list
• Echa un vistazo a la lista = Take a look at the list / Have a quick look at the list

Mira la lista is more direct and simple. Echa un vistazo a la lista sounds a bit more idiomatic and often a little softer or more conversational.

Both are very common and natural.

Yes. In everyday Spanish, you will often hear:

• Échale un vistazo a la lista

This is very common and natural. The le does not really change the basic meaning; it is just a common feature of how this expression is often used.

So both of these are fine:

• Echa un vistazo a la lista
• Échale un vistazo a la lista

For a learner, the version without le is often easier to analyse, but you should definitely recognise both.

No. They sound the same, but they are different words.

• echa comes from echar
• hecha comes from hacer and means done or made in the feminine form

Examples:

• Echa un vistazo = from echar
• La tarea está hecha = from hacer

This is a very common spelling confusion for learners, because the h is silent.

Yes, very common. It is a normal, everyday way to say have a look or take a quick look in Spain.

It sounds natural in speech and writing, and it is a very useful expression to remember as a fixed phrase.