Io insegno a mio cugino a suonare il pianoforte.

Yes—Italian usually does drop them, because the verb ending already shows the subject.

• insegno = I teach
• so Io insegno and Insegno both mean I teach

Adding io gives extra emphasis or contrast, for example:

• Io insegno, tu impari. = I teach, you learn.

In a neutral sentence, most Italians would simply say:

• Insegno a mio cugino a suonare il pianoforte.

Because mio cugino is the person receiving the teaching, so it works like an indirect object here.

With insegnare, Italian commonly uses this pattern:

• insegnare qualcosa a qualcuno = to teach something to someone
• insegnare a qualcuno a fare qualcosa = to teach someone to do something

So in your sentence:

• a mio cugino = to my cousin

That first a marks the person you are teaching.

That a belongs to a different structure.

Here the pattern is:

• insegnare a qualcuno a fare qualcosa

So:

• a mio cugino = to my cousin
• a suonare = to play

The second a introduces the action being learned. In English, this is the to in teach someone to play.

Because the two languages build the sentence differently.

English:

• I teach my cousin to play the piano.

Italian:

• Insegno a mio cugino a suonare il pianoforte.

The key difference is that Italian marks the person with a:

• my cousin in English
• a mio cugino in Italian

So even though both languages have something like to, they are not lined up word-for-word.

With singular close family members, Italian often uses the possessive without the article.

Examples:

• mio padre
• mia madre
• mio fratello
• mia sorella
• mio cugino

So a mio cugino is normal.

But there are important limits:

• If the family noun is plural, you usually use the article: i miei cugini
• If the family noun is modified, the article often appears: il mio caro cugino
• Some words, especially less “close-family” sounding ones or altered forms, may also take the article

Insegno is the first person singular, present tense of insegnare.

So:

• insegnare = to teach
• insegno = I teach / I am teaching (depending on context)

Italian present tense often covers both:

• habitual actions: I teach
• actions happening now: I am teaching

In Italian, musical instruments usually take the definite article after suonare.

So you normally say:

• suonare il pianoforte
• suonare la chitarra
• suonare il violino

This is very natural in Italian. Even though English and Italian both happen to use the with piano, the Italian rule is broader and more consistent with instruments.

Yes. Pianoforte is perfectly correct, but piano is very common in everyday speech.

So these both work:

• Insegno a mio cugino a suonare il pianoforte.
• Insegno a mio cugino a suonare il piano.

The second one sounds a bit more conversational.

Yes. You can replace a mio cugino with the indirect object pronoun gli.

So:

• Insegno a mio cugino a suonare il pianoforte. becomes
• Gli insegno a suonare il pianoforte.

Here:

• gli = to him

If the person were female:

• Le insegno a suonare il pianoforte.

Not completely. Italian word order is flexible, although some versions sound more neutral than others.

The most neutral form is:

• Insegno a mio cugino a suonare il pianoforte.

You can also say:

• Io insegno a mio cugino a suonare il pianoforte.
  if you want emphasis on I

Other word orders are possible in the right context, but this standard order is the safest for learners.

Yes, if what you teach is a noun rather than an action.

Compare:

• Insegno l’italiano a mio cugino. = I teach Italian to my cousin.
• Insegno a mio cugino a suonare il pianoforte. = I teach my cousin to play the piano.

So:

• teach something to someone
  → insegnare qualcosa a qualcuno

• teach someone to do something
  → insegnare a qualcuno a fare qualcosa

That is why your sentence needs both uses of a.