1.- Puedes decir "El perro murió". Pero "El perro se me murió" muestra que quien lo dice tiene un cierto apego o cariño hacia el perro, muestra un sentimiento hacia él, lo considera algo suyo. En este caso "se" no tiene función sintáctica. 2.- "Se me acaba de ocurrir una idea", es equivalente a "he tenido una idea" (I've had an idea). Si dices "Me acaba de ocurrir una idea", es algo que te ha sucedido (An idea happens to me <- this has no sense).

1. "El perro murió" is the standard, but sounds a bit bookish and cold. "El perro se murió" turns the verb into a reflexive verb by using "se", and thus sounds more natural. Literally, it would be translated as "he died himself", because "se" has the same value as "himself". "me", "te" and "se" and "nos" are used to transform verbs into reflexive ones, and literally mean "myself", "yourself", "himself/herself/themselves" and "ourselves". In Spanish, it's very common to turn a verb into a reflexive verb when there's no direct object. For example, in English, you say "I sit", in Spanish we say "(Yo) me siento". More examples: "(Yo) me voy" (I go) "(Yo) me quedo" (I stay) "El perro se me murió", as Javier pointed out, adds the "me" to add the meaning that he didn't just "die himself", he "died (himself) on me". You might have heard "Don't die on me!" in American movies. In Spanish, it would be almost literally translated, "No te me mueras", just adding the reflexive "te".

"El perro se murió" This sentence means that ANY dog was died But... "El perro se me murió" This sentence means that YOUR dog was died = "Mi perro se murió" The verb "morir + se" indicates a reflexive verb: Yo me muero / tú te mueres / Él. Ella. Usted se muere / Nosotros nos morimos / Vosotros os morís / Ellos. Ellas. Ustedes se mueren. "Se me acaba de ocurrir una idea" In this case the "se" particle is working as a function of emphasizing the Subject thus, the speaker (Yo) It hasn´t any sintactic function (as my colleage said) "me" is working as Indirect Object replacing the pronoum "a mí (Yo)" The verb "ocurrir + se = pronominal verb" in this context, it means to have an idea. Se me ha ocurrido una gran idea => I've got a great idea. Se me ocurre que podemos ir al cine => It occurs to me that we can go to the cinema. If you have any question, please let me know. Regards. Antonio