English:
Formal: Dear Ms. Smith; informal: Dear Paul, both of which are clear enough.
Fml: Yours truly, or Sincerely yours, or Yours - all descendents of “I beg to remain truly (or Sincerely) your most obedient servant.”
Infml: Love, or Cheers,
French:
Fml: Chère Mme Desautels; infml: Cher Henri (“dear” again).
Fml: Veuillez (or Je vous prie d’)agréer l’expression de mes sentiments les meilleurs (or distinguées): Please permit the expression of my best wishes.
Infml: Hmm… You know, I’ve never had occasion to send an informal letter in French (I was going to say “French letter”, but I thought better of it). I’d ask my roommate, but he’s asleep.
Spanish:
Fml:
For women: Estimada señor(it)a (Esteemed lady).
For men: Muy señor mío (An odd sentence… “señor” means “Mr.” or “sir” or “gentleman”, of course, but here it’s an adjectival noun… “My distant superior”.)
Infml: Querido Juan or Querida Pilar (“dear” again).
Fml:
Atentamente, (Attentively,)
Infml:
Reciba un cordial saludo de, (Receive cordial greetings from,)
Esperanto:
Fml: Estimata sinjor(in)o or Estimataj gesinjoroj (Dear Sir/Dear Madam/Ladies and gentlemen)
Infml: Kara Kàlmàn or Karaj amikoj (“dear” again)
As for the closing, some different ones include:
Amike salutas, (the most common informal one: “… sends friendly salutations”)
Kore, (“from the heart”)
Dankon, (“thank you”)
Samideane, (“as a fellow member of the Esperanto movement”)
Or you can get as flowery as you want:
Atendas avide vian respondon, kaj dankas vin antaue,
("…eagerly awaits your response, and thanks you ahead of time.")
Other languages please, with translations?
Y’r most humble and ob’t svt,
Matt