Mi prima va al trabajo en moto y siempre lleva casco.

No. Mi does not change for gender.

• mi primo = my male cousin
• mi prima = my female cousin

It only changes for number:

• mi prima = my cousin
• mis primas = my cousins

So mi works with both masculine and feminine singular nouns.

Prima means female cousin.

Spanish distinguishes:

• primo = male cousin
• prima = female cousin

So mi prima specifically means my female cousin.

Because ir is the infinitive form, meaning to go, while va is the conjugated form used with mi prima.

Here, mi prima is she, so you need the third person singular of ir:

• yo voy = I go
• tú vas = you go
• él/ella va = he/she goes

So:

• Mi prima va... = My cousin goes...

Because in Spanish, a + el contracts to al.

So:

• a + el trabajo → al trabajo

This contraction is mandatory.

Examples:

• Voy al cine = I go to the cinema
• Va al trabajo = She goes to work

The only common exception is when el is part of a proper name:

• Voy a El Escorial
  not al Escorial

Both can often be translated as goes to work, but they are not exactly the same.

• va al trabajo = she goes to her workplace / to work
• va a trabajar = she goes to work / in order to work

So:

• al trabajo focuses more on the destination
• a trabajar focuses more on the activity

In this sentence, va al trabajo is talking about her usual trip to her workplace.

Because Spanish normally uses en to talk about means of transport.

Examples:

• en coche = by car
• en tren = by train
• en autobús = by bus
• en moto = by motorbike / on a motorbike

Even though English says on a motorbike, Spanish uses en, not something equivalent to on here.

With common transport expressions, Spanish usually leaves out the article.

So you normally say:

• en coche
• en tren
• en metro
• en moto

Not usually:

• en la moto
  unless you mean on the motorbike in a more specific sense, for example on that particular motorbike

So in this sentence, en moto means by motorbike / on a motorbike in a general way.

Yes. Moto is the very common everyday short form of motocicleta.

In Spain, moto is the normal word people use in conversation.

So:

• en moto sounds natural and everyday
• en motocicleta is correct, but less common in ordinary speech

Yes, llevar can mean to carry or to take, but it also very often means to wear.

In this sentence:

• lleva casco = she wears a helmet

This is very natural Spanish.

Some examples:

• Lleva gafas = She wears glasses
• Lleva chaqueta = He is wearing a jacket
• Siempre lleva casco = She always wears a helmet

So llevar often means to have something on your body / to be wearing it.

Because Spanish often leaves out the article with clothing and similar items when speaking generally.

So:

• lleva casco = she wears a helmet / she wears helmet protection in general

If you say:

• lleva un casco = she is wearing a helmet
  (one helmet, less general)
• lleva el casco = she is wearing the helmet
  (a specific helmet)

In your sentence, lleva casco sounds general and natural: it describes her usual habit.

Because adverbs like siempre often go before the conjugated verb in Spanish.

So:

• siempre lleva casco = she always wears a helmet

This is the most neutral and common order.

Spanish word order is fairly flexible, so you may also hear:

• lleva siempre casco

But siempre lleva casco is very standard and natural.

Yes. Spanish, like English, often uses the present tense for habitual actions.

So this sentence means that this is something she normally does:

• Mi prima va al trabajo en moto = My cousin goes to work by motorbike
• siempre lleva casco = and she always wears a helmet

It is not necessarily happening right now. It describes her regular routine.

Yes. That would also be correct and natural.

Both are possible:

• Mi prima va al trabajo en moto
• Mi prima va en moto al trabajo

The difference is mainly focus:

• va al trabajo en moto emphasizes the destination first, then the means of transport
• va en moto al trabajo brings en moto a little more forward

Both are good Spanish.