Per Capodanno, mi firmi qui, per favore, così confermiamo la prenotazione.

Per Capodanno means for New Year’s / for the New Year period. Here per suggests purpose or reference: the reservation is for that occasion.

By contrast, a Capodanno usually means on New Year’s Day / at New Year as a point in time.

So in this sentence:

• Per Capodanno = the booking concerns New Year’s
• A Capodanno = something happens on New Year’s

Because the sentence is about confirming a reservation connected to that holiday period, per is the natural choice.

Usually, no. Italian holiday names often appear without an article in this kind of expression:

• a Natale
• a Pasqua
• a Capodanno

and also:

• per Natale
• per Capodanno

So Per Capodanno is perfectly normal.

Capodanno literally means New Year’s Day, but in everyday use it can also refer more broadly to the New Year period / New Year’s celebrations.

In a sentence about a reservation, it often means something like:

• for New Year’s
• for the New Year holiday period

So it does not have to mean only the single day of January 1st.

No, it does not mean sign me.

Here mi is an indirect object pronoun, and it means something like:

• for me
• for us / on my side
• so I have your signature

So mi firmi qui is roughly:

• sign here for me, please
• would you sign here, please

In service situations, Italian often uses this kind of mi even where English would leave it out.

Yes. Here firmi is the formal imperative of firmare, used with Lei.

Italian uses the present subjunctive form to make the formal imperative:

• firma! = sign! (informal, tu)
• firmi! = sign! (formal, Lei)

So mi firmi qui, per favore is a polite request to one person addressed formally, like a customer.

Both, in a way.

Formally speaking:

• the shape of the verb is the present subjunctive
• its function here is imperative because it gives a polite command/request

So in this sentence, it is best understood as the formal imperative, even though it looks like a subjunctive form.

That is true for the informal affirmative imperative, but not for the formal imperative.

Compare:

• Firmami qui. = Sign here for me. (informal tu)
• Mi firmi qui. = Sign here for me. (formal Lei)

So:

• with tu affirmative commands, pronouns usually attach to the end
• with Lei formal commands, pronouns go before the verb

That is why mi firmi is correct here.

Yes, but it would change the register.

• Firma qui = informal command to tu
• Firmi qui = formal command to Lei

So if you are speaking to a customer politely, firmi is more appropriate.

A very common polite alternative in speech is also:

• Mi firma qui, per favore?

That uses the present indicative as a polite question/request instead of the formal imperative.

Yes. Firmare means to sign, especially to write your signature on a document, form, receipt, contract, and so on.

Examples:

• firmare un contratto = to sign a contract
• firmare qui = to sign here

So in this sentence, the speaker is asking someone to sign in a specific place.

Here così means something like:

• so
• that way
• thus

It introduces the result or purpose of the signature:

• mi firmi qui, per favore, così confermiamo la prenotazione
• sign here, please, so that we can confirm the reservation

It is very common in spoken Italian to use così in this way.

Confermiamo means we confirm.

The speaker is presenting the action as something done by we:

• the staff and the system
• the business side
• or the speaker together with the customer in a practical sense

Italian often uses we in customer-service language where English might prefer:

• so we can confirm the booking
• so the booking can be confirmed

So confermiamo sounds natural in this context.

It can mean either, depending on context. Italian present tense often covers both:

• we confirm
• we are confirming

Here the idea is basically:

• once you sign, we can confirm the reservation

So English usually translates it with we confirm or we can confirm depending on what sounds most natural.

Prenotazione can mean both reservation and booking, depending on context.

For example:

• hotel = reservation / booking
• restaurant = reservation
• doctor’s appointment or ticketed event = sometimes also booking

So la prenotazione here is simply the reserved booking already being processed.

Yes, it can move. Per favore is flexible.

All of these are possible:

• Mi firmi qui, per favore.
• Per favore, mi firmi qui.
• Mi firmi, per favore, qui.

The version in your sentence sounds natural because per favore softens the request right after the command.

Yes, it is natural, especially in a formal service context such as a hotel, agency, or reception desk.

It sounds polite and professional. A native speaker might also say similar things like:

• Mi firmi qui, per favore, così possiamo confermare la prenotazione.
• Mi firma qui, per favore, così confermiamo la prenotazione.

Your original sentence is fully idiomatic.