Mi padre me pasa el martillo porque uno de los clavos es demasiado pequeño.

Here, pasar means to pass or to hand something to someone. So Mi padre me pasa el martillo means My father passes/hand me the hammer.

A common confusion is that pasar can also mean to happen, but not in this sentence. Here it has a direct object, el martillo, so the meaning is clearly to pass/hand over.

Me is the indirect object pronoun meaning to me.

So:

• Mi padre = my father
• me = to me
• pasa = passes / hands
• el martillo = the hammer

Spanish usually uses these short object pronouns before the conjugated verb:

• me = to me
• te = to you
• le = to him/her/you
• nos = to us
• os = to you all
• les = to them/you all

Normally, no. In ordinary Spanish, the indirect object pronoun is the normal way to say to me here, so me pasa is the natural structure.

You can add a mí for emphasis or contrast, but usually together with me, not instead of it:

• Mi padre me pasa el martillo a mí. = My father passes the hammer to me.

That sounds emphatic, as if you are contrasting me with someone else.

Spanish uses articles very often, more often than English does. In this sentence, el martillo refers to a specific hammer, probably one already understood from the situation.

So Spanish naturally says:

• el martillo = the hammer
• los clavos = the nails

Even where English might sometimes omit an article, Spanish often keeps it.

Because porque and por qué are different words with different jobs:

• porque = because
• por qué = why

So in this sentence:

• ...porque uno de los clavos es demasiado pequeño.
  = ...because one of the nails is too small.

If you were asking a question, you would use por qué:

• ¿Por qué me pasa el martillo? = Why is he handing me the hammer?

The structure uno de + plural noun means one of the ...

So:

• uno de los clavos = one of the nails

You cannot say un de los clavos.

This is because uno is being used in the partitive expression one of the group. Compare:

• un clavo = a nail
• uno de los clavos = one of the nails

Because the meaning is one out of a group of nails.

In uno de los clavos:

• uno = one
• de los clavos = of the nails

So the noun after de stays plural because it refers to the whole set. English works the same way:

• one of the nails
• not one of the nail

Because the subject is uno, which here stands for one nail. Since clavo is masculine singular, the adjective must also be masculine singular:

• pequeño = masculine singular
• pequeña = feminine singular
• pequeños = masculine plural
• pequeñas = feminine plural

So:

• uno de los clavos es demasiado pequeño
• una de las tablas es demasiado pequeña

The adjective agrees with the thing being described.

In this sentence, demasiado is an adverb meaning too or overly, and adverbs do not agree in gender or number.

So:

• demasiado pequeño = too small
• demasiado grande = too big
• demasiado pequeña = too small

Notice that demasiado stays the same there.

But demasiado can also be an adjective or determiner, and then it does change:

• demasiados clavos = too many nails
• demasiada fuerza = too much force

So the form depends on its job in the sentence.

Spanish often uses the simple present where English might use either the simple present or the present continuous, depending on context.

So Mi padre me pasa el martillo can describe a present action naturally in Spanish.

You could also say:

• Mi padre me está pasando el martillo

That emphasizes the action as ongoing right now. But the simple present is very normal and often the default choice.

Yes, in many contexts you could say dar, but the nuance changes a little.

• dar = to give
• pasar = to pass / hand over

So:

• Mi padre me da el martillo = My father gives me the hammer
• Mi padre me pasa el martillo = My father passes/hand me the hammer

Pasar feels more like physically handing something across, especially in a shared space.

Clavo means a metal nail, the kind you use with a hammer.

A fingernail or toenail is uña.

So in this sentence:

• martillo
  • clavo clearly go together as hammer
    • nail

That helps confirm the meaning.