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You can use them as you want, there's no real difference. Just in the "la semana reciente", at least here in mexico it's not really usual to said that one, it's more natural just "la semana pasada"
Also it would be more natural "la proxima semana" instead of "la semana proxima"
As for the other question: Hoy es 18 de febrero, si digo"el próximo 30 de mayo " ¿ es de este año o del año que viene?
It would be of this year!! The only way it would be assumed that it's until the next year would be if the date is already gone, for example if i say that something is going to happen in january it would be assumed that it's until january next year (since we're already in february)
Actually we would't use "el proximo 30 de mayo", in a casual conversation just saying "el 30 de mayo" would be perfectly understood!!
