What is the ruling on using the phrase "And to you our greetings" or "And to you our greeting" at the end of official letters, considering that "greetings are for Allah alone, without partner"?

There is no harm in saying: "My greetings to you" or "With our greetings." The greetings that are exclusively for Allah alone are the complete, general greetings as in the Tashahhud (definition). As for a specific greeting from one person to another, there is no objection to it. Sheikh Al-Uthaymeen cited as evidence the verse in which Allah, the Most High, says: (And when you are greeted with a greeting, greet in return with one better than it, or [at least] return it [in a similar manner]) [An-Nisa/86], emphasizing the permissibility of these phrases as long as they are not taken as a substitute for the prescribed Islamic greeting (Salam).