No, you don’t have to leave the country. The UK government travel advice for Turkey states:
If you plan to stay in Turkey longer than 90 days, you must apply for a residence permit before the 90 days ends or apply for a longer-stay visa before you travel
So as long as you apply for it before your visa expires you can stay in the country.
From https://embamex.sre.gob.mx/emiratosarabesunidos/index.php/es/2016-05-07-10-39-06 the documents needed to apply for a visitor visa are:
1.- Visa application fully filled with capital letters.
2.- Valid passport with at least two blank pages. Validity of at least six months prior to expiration.
3.- Passport photocopies of the first 3 pages and all ...
Yes, you can apply for a tourist visa for Mexico in the UAE if you are a resident there.
On the list of requirements on the consulate of Mexico in the UAE website:
The applicant has to be a resident in the United Arab Emirates
So as long as you are a legal resident in the UAE you will be fine.
No, this is not possible as as the receipt only proves your legal status in Italy. you are only allowed to enter and exit the Schengen area through Italy.
Verified by the Polizia di Stato website:
Foreigners awaiting renewal of their residence permits can leave and re-enter Italy if they hold:
the receipt issued by Italian Post offices (Poste ...