Appoggio il mento sulla mano quando penso.

Italian often leaves out the subject pronoun when it is clear from the verb ending.

• appoggio = I rest / I place / I lean
• penso = I think

Because the -o ending already shows I, io is usually unnecessary.
You can say Io appoggio... if you want extra emphasis, contrast, or clarity, but the version without io is more natural in ordinary speech.

Here, appoggio comes from appoggiare and means something like:

• I rest
• I lean
• I place

So Appoggio il mento sulla mano means I rest my chin on my hand.

This verb is very common for putting or resting something against/on something else:

• Appoggio il libro sul tavolo = I put/rest the book on the table
• Appoggio la testa al muro = I lean/rest my head against the wall

In this sentence, it suggests a physical pose: the chin is supported by the hand.

Italian often uses the definite article with body parts where English prefers a possessive.

So instead of saying:

• my chin
• my hand

Italian often says:

• il mento
• la mano

This is especially common when it is already obvious whose body part it is.

So:

• Appoggio il mento sulla mano
  literally = I rest the chin on the hand
  but naturally = I rest my chin on my hand

This is a very common Italian pattern:

• Mi lavo le mani = I wash my hands
• Alzo la testa = I raise my head

Sulla means on the and is the natural choice here because the chin is resting on top of / against the hand.

• su + la = sulla

So:

• sulla mano = on the hand

Other options would mean something different:

• nella mano = in the hand
  This would suggest something inside the hand, like holding an object.
• alla mano is not natural here for physical contact of this kind.

So Appoggio il mento sulla mano is the normal way to say I rest my chin on my hand.

Yes, mano is singular because the sentence describes the chin resting on one hand.

• sulla mano = on my hand
• sulle mani = on my hands

If someone used both hands, Italian could say:

• Appoggio il mento sulle mani = I rest my chin on my hands

So the singular here simply matches the image being described.

Quando penso means when I think.

It is a time clause introduced by quando = when.

• quando = when
• penso = I think

So the whole sentence means something like:

• I rest my chin on my hand when I think
• When I think, I rest my chin on my hand

The present tense in Italian works well here for a habitual action, just like in English.

Most naturally, it describes a habit or something you typically do.

• Appoggio il mento sulla mano quando penso
  = I rest my chin on my hand when I think

Italian present tense can express:

1. a habitual action
   
   • Bevo caffè ogni mattina = I drink coffee every morning
2. something happening now, depending on context
   
   • Mangio = I’m eating / I eat

In this sentence, because of quando penso, the habitual meaning is the most natural: this is something you tend to do while thinking.

You could, but appoggio is better here.

• mettere = to put
• appoggiare = to rest / lean / place against

Metto il mento sulla mano is understandable, but it sounds more like simply putting the chin there.
Appoggio better captures the idea of resting or supporting the chin on the hand, which is the natural meaning in this context.

So appoggio is the more idiomatic choice.

Yes, both are correct.

• Appoggio il mento sulla mano quando penso
• Quando penso, appoggio il mento sulla mano

The difference is mainly emphasis and flow:

• Starting with Appoggio... puts the focus first on the action.
• Starting with Quando penso... puts the time/situation first: When I think...

Italian word order is fairly flexible, especially with time expressions and subordinate clauses like quando penso.

Because appoggiare does not need a reflexive pronoun in this sentence.

You are directly acting on a body part:

• appoggio il mento = I rest my chin

Italian often does this without mi, using the article with the body part instead:

• Alzo la mano = I raise my hand
• Chiudo gli occhi = I close my eyes

A reflexive form would be used in different structures, but here the simple non-reflexive verb is the normal choice.

Yes. Mano is an important irregular noun:

• la mano = the hand
• le mani = the hands

So even though it ends in -o, it is feminine.
That is why the sentence uses:

• sulla mano
  not sullo mano

Because:

• su + la = sulla

This is a common word to memorize as an exception.

Yes. The verb appoggiare can mean to support, depending on context.

For example:

• Appoggio la tua idea = I support your idea

But in this sentence, the meaning is clearly physical, not abstract:

• Appoggio il mento sulla mano = I rest my chin on my hand

So the exact English translation depends on context.

In Italian, many prepositions combine with the definite article.

Here:

• su = on
• la = the
• su + la = sulla

So sulla mano literally means on the hand.

Other similar combinations are:

• sul tavolo = on the table (su + il)
• sui libri = on the books (su + i)
• sulle sedie = on the chairs (su + le)

These combined forms are extremely common and important to learn early.