Appoggio il gomito sul tavolo mentre leggo.

Appoggio is the present indicative, and in Italian it often covers both:

• I rest / I place
• I am resting / I am placing

So Appoggio il gomito sul tavolo mentre leggo can naturally mean I rest my elbow on the table while I read.

Italian does have a progressive form (sto appoggiando), but it is used less often than English am ...-ing. In a sentence like this, the simple present sounds perfectly normal.

Here appoggiare means to rest, to place, or to lean something on a surface.

In this sentence, appoggio il gomito sul tavolo means that the speaker is resting their elbow on the table.

A few related ideas:

• appoggiare qualcosa su... = to place/rest something on...
• appoggiarsi a / su... = to lean on...

So this sentence uses the transitive verb: the speaker is doing something to il gomito.

Italian often uses the definite article (il, la, i, le) where English uses a possessive like my, your, his—especially with body parts.

So:

• Appoggio il gomito sul tavolo
  literally looks like I rest the elbow on the table,
  but it naturally means I rest my elbow on the table.

Because the subject is clear, Italians usually do not need to say mio here.

Compare:

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

Sul means on the and is a combination of:

• su = on
• il = the

So:

• su + il = sul

You need the article because tavolo here is a specific noun phrase: the table.

Examples:

• sul tavolo = on the table
• sulla sedia = on the chair
• sui libri = on the books

Nel tavolo would mean in the table, which does not fit this meaning.

Mentre means while.

It connects two actions that happen at the same time:

• Appoggio il gomito sul tavolo
• leggo

So the idea is: I rest my elbow on the table while I read.

It is a very common way to express simultaneous actions in Italian.

Leggo is the first-person singular present of leggere (to read).

Because the subject is I, the verb must be conjugated:

• io leggo = I read / I am reading

Why not the others?

• leggere = infinitive, to read
• leggendo = gerund, reading

After mentre, Italian normally uses a conjugated verb:

• mentre leggo = while I read / while I am reading

Italian usually drops subject pronouns when the verb ending already shows who the subject is.

Here:

• appoggio clearly means I rest
• leggo clearly means I read

So io is not necessary.

You could say:

• Io appoggio il gomito sul tavolo mentre leggo

But that would usually add emphasis, contrast, or clarity.

It can be either, depending on context.

The Italian present tense is flexible. It can describe:

• a habit: I rest my elbow on the table while I read
• an action happening now: I’m resting my elbow on the table while I read

Without more context, both are possible.

Yes, grammatically you can, but it is usually less natural unless you want special emphasis.

Normally Italian prefers:

• appoggio il gomito sul tavolo

Using il mio gomito may sound more emphatic, contrastive, or unusually explicit, for example if you are contrasting it with someone else’s elbow.

Yes, they are related but not the same.

• Appoggio il gomito sul tavolo = I rest my elbow on the table
• Mi appoggio al tavolo = I lean on the table

The first sentence focuses on the elbow as the thing being placed on the table.
The second focuses on you yourself leaning against or onto the table.

So appoggiare and appoggiarsi are connected, but they are used differently.

Because il gomito and il tavolo are nouns, and Italian usually uses an article with singular countable nouns in this kind of sentence.

But mentre leggo is not a noun phrase. It is a clause:

• mentre = while
• leggo = I read

Articles go with nouns, not with clauses like this.

The basic structure is:

• Appoggio = verb
• il gomito = direct object
• sul tavolo = prepositional phrase
• mentre leggo = time clause

So literally:

• [I rest] [the elbow] [on the table] [while I read]

This word order is natural in Italian. You could rearrange parts for emphasis, but the given version is straightforward and idiomatic.