Tus lápices, cuyos colores son vivos, están sobre la mesa.

Because lápices is plural, the possessive adjective also has to be plural.

• tu = your, before a singular noun
• tus = your, before a plural noun

So:

• tu lápiz = your pencil
• tus lápices = your pencils

Spanish possessive adjectives agree with the thing possessed, not with the owner.

The singular form is lápiz. When it becomes plural, Spanish changes z to c before -es:

• lápiz → lápices

The accent mark stays because the stress remains on the first syllable: LÁ-pi-ces.

This is a very common spelling change in Spanish:

• luz → luces
• pez → peces

Cuyos means whose.

In this sentence, it connects lápices with colores:

• Tus lápices, cuyos colores son vivos...
• Your pencils, whose colors are vivid...

It shows possession or relationship. The colors belong to the pencils.

This word is often one of the trickier relative pronouns for English speakers because English uses whose very broadly, but Spanish uses forms of cuyo that must agree in gender and number with the thing possessed.

Because cuyo agrees with the noun that comes after it, not with the noun before it.

Here the noun after it is colores, which is:

• masculine
• plural

So the correct form is cuyos.

The full set is:

• cuyo = masculine singular
• cuya = feminine singular
• cuyos = masculine plural
• cuyas = feminine plural

Examples:

• el libro cuyo autor... = the book whose author...
• la casa cuya puerta... = the house whose door...
• los lápices cuyos colores... = the pencils whose colors...
• las mesas cuyas patas... = the tables whose legs...

It is completely correct, but it often sounds somewhat formal or written.

In everyday speech, many speakers prefer alternatives such as:

• Tus lápices, que tienen colores vivos, están sobre la mesa.
• Tus lápices, con colores vivos, están sobre la mesa.

So cuyo is useful and important to know, especially for reading and more careful writing, but it may sound a bit elevated in casual conversation.

Because that part is extra, descriptive information about tus lápices. It is a nonrestrictive relative clause.

The sentence is basically:

• Tus lápices están sobre la mesa.

Then the speaker adds extra information:

• cuyos colores son vivos

So the commas show that this part is not essential for identifying which pencils we mean. It is just additional description.

In English, this works similarly:

• Your pencils, whose colors are vivid, are on the table.

Because vivos describes colores, and colores is masculine plural.

Adjectives in Spanish usually agree with the noun they describe:

• color vivo = vivid color
• colores vivos = vivid colors

It does not agree with lápices here; it agrees directly with colores.

Not in this context. Vivo can mean different things depending on the situation.

With colors, vivo means:

• vivid
• bright
• intense

So colores vivos means bright/vivid colors, not alive colors.

This is a very normal use in Spanish.

Because Spanish uses ser and estar for different kinds of meanings.

• son vivos uses ser because it describes a characteristic of the colors
• están sobre la mesa uses estar because it gives the location of the pencils

So:

• Los colores son vivos = The colors are vivid
• Los lápices están sobre la mesa = The pencils are on the table

This is a classic ser vs. estar distinction:

• ser for identity/characteristics
• estar for location/state

Because the subject is plural: Tus lápices.

So the verb has to be plural too:

• Tu lápiz está... = Your pencil is...
• Tus lápices están... = Your pencils are...

Even though there is a long clause in the middle, the main subject is still Tus lápices.

Sobre la mesa means on the table or on top of the table.

That is the most natural choice if the pencils are physically resting on the surface.

• sobre la mesa = on the table
• en la mesa can sometimes mean on the table in a broader sense, but it may sound less precise depending on context

In many cases both are possible, but sobre clearly emphasizes position on top of the surface.

Yes, but not unusually so. The basic structure is:

• Tus lápices = subject
• cuyos colores son vivos = descriptive relative clause
• están sobre la mesa = main verb + location

If we remove the relative clause, the sentence becomes very straightforward:

• Tus lápices están sobre la mesa.

The middle part simply interrupts the main sentence to add more information.

Yes. That is a very natural alternative, especially in conversation.

It means nearly the same thing, but it is less formal than cuyos.

Compare:

• Tus lápices, cuyos colores son vivos, están sobre la mesa.
  More formal, more literary/written.

• Tus lápices, que tienen colores vivos, están sobre la mesa.
  More common in everyday speech.

So if your goal is natural spoken Latin American Spanish, the second version may sound more conversational.