Is it legal for me to put stickers and writing on payphones?

No. Payphones are privately owned products licensed by the city in which they are located. Payphones are not public property, nor are they owned by vast corporate entities.

Vandalism of public phones typically results in the owner of the phone being issued a penalty.

New York City collects about $1,000,000 a year enforcing fines involving graffiti and vandalism of payphones. Other municipalities earn significant income from these penalties as well. If you enjoy robbing independent payphone providers of enough money to put them out of business and shuffling that money to local governments then enjoy your bad self.