Hi Tommaso. This is the present of the verb "essere" (to be):
Italian is a pro-drop language, it means that pronouns may be generally omitted. "Sono" is used both for 1st person singular and 3rd person plural (not for "we" but for "they"). However, in your example the adjective should be set accordingly to the subject, so "I am happy" is "sono felice", while "they are happy" is "sono felici" (so there is no ambiguity). The verb "essere" is an irregular one, normally the forms for the two persons are different.
As for your second question (the correct sentence is "Lei farà"), the third person feminine singular pronoun, i.e. "Lei" (she), is used as a polite form in place of "tu" (you), the same role that in French can be played by the "vous" pronoun. For example, if you go in a bar and you want a coffee, to be polite towards the bartender you should say: "Mi può fare un caffè?", which is literally translated as: "Can she make a coffee for me?".
Hope this helps.