A veces uso cinta para cerrar la caja y otras veces uso pegamento.

Spanish often drops subject pronouns when they are not needed, because the verb ending already shows who is doing the action.

• uso = I use
• so yo uso and uso both mean the same thing here

Including yo is possible, but it usually adds emphasis or contrast. In a neutral sentence, Spanish normally just says uso.

A veces is a fixed expression meaning sometimes.

It uses veces, the plural of vez (time / occasion), because the idea is at times or on some occasions. You should learn a veces as a whole expression.

Examples:

• A veces leo por la noche.
• A veces no tengo hambre.

Otras veces means other times or at other times.

In this sentence, the pattern is:

• A veces...
• y otras veces...

This is a very common way to contrast two habits or repeated possibilities:

• Sometimes I use tape, and other times I use glue.

So otras veces works naturally as the second half of that contrast.

Para + infinitive is a very common Spanish structure used to express purpose — in other words, in order to do something.

So:

• uso cinta para cerrar la caja
  = I use tape to close the box / in order to close the box

Here:

• para = for / in order to
• cerrar = infinitive, to close

This structure is extremely common:

• Uso un cuchillo para cortar el pan.
• Estudio para aprender.

Spanish uses articles (el, la, los, las, un, una...) more often than English does.

Here, la caja sounds natural because Spanish usually prefers an article with a specific noun in a sentence like this. Even if English might say the box or sometimes just imply it, Spanish generally includes the article.

So:

• cerrar la caja = to close the box

Leaving out la would sound incomplete in standard Spanish.

Because pegamento is being used as a material or substance in a general sense, not as one specific glue container.

Compare:

• Uso pegamento. = I use glue.
• Uso un pegamento especial. = I use a special glue.

This is similar to English, where we often say I use glue, not I use a glue, unless we mean a particular type.

The same idea can apply to other materials:

• Uso papel
• Uso agua
• Uso pintura

It can be understood, but in Spain people often say:

• cinta adhesiva = adhesive tape
• celo = sticky tape / sellotape

So cinta by itself may be understandable from context, but if you want to be more natural in Spain, cinta adhesiva or celo may be more common depending on the situation.

For example:

• Uso celo para cerrar la caja.
• Uso cinta adhesiva para cerrar la caja.

The normal word for and is y.

Spanish changes y to e only before words that begin with an i sound:

• padre e hijo
• España e Italia

But otras begins with an o sound, so it stays y:

• y otras veces

Yes. That is also correct, but it means something slightly different in structure.

• Uso cinta para cerrar la caja
  focuses on what I use and the purpose
• Cierro la caja con cinta
  focuses on how I close the box

So both are natural, but they are built differently:

• usar algo para + infinitive
• hacer algo con + noun

Examples:

• Uso pegamento para pegar el papel.
• Pego el papel con pegamento.

• cinta is feminine: la cinta
• caja is feminine: la caja
• pegamento is masculine: el pegamento

That matters for articles and adjectives:

• la cinta blanca
• la caja grande
• el pegamento fuerte

Yes, Spanish allows some flexibility in word order, especially with time expressions like a veces and otras veces.

For example, these are possible:

• A veces uso cinta para cerrar la caja y otras veces uso pegamento.
• Uso cinta a veces para cerrar la caja y otras veces uso pegamento.
• Otras veces uso pegamento.

But the original version is the most natural and clear for this kind of contrast:

• A veces... y otras veces...

It sounds balanced and idiomatic.