When I started learning Portuguese years ago, all the books I used at the time told me that when using possessive adjectives you also have to put the corresponding definite article in front of the possessive adjective. For instance, "my name" should be translated as "o meu nome"; "his house" as "a sua casa" etc. Is this rule still in practice in Portuguese? And is there a difference between European and Brazilian Portuguese in using the definite article with possessive adjectives?
The rule is different. It only applies when the possessive pronoun is substantive:
Este é meu livro, o seu é o outro. (This is my book, yours is the other one.)
In most other contexts, the use of the definite article is optional, not mandatory, and I don't think there is any difference between Brazilian Portuguese and European Portuguese in this regard. In this it is actually different from English, Spanish or French, where the possessive pronoun excludes the article:
*The my house
*La mi casa
*Le mon frère
To demonstrate that the article is not mandatory in other situations, here is a list of books beggining with "M"; it features titles such as
Meu amigo canguru (Pinto, Ziraldo Alves) (My friend the Kangaroo)
Meu caro H (Thomaz, Samir) (My dear H)
Meu corpo (Finifter, Germaine) (My body)
Meu Filho tem medo (Schachter, Robert) (My son is afraid)
Meu livro de ciências (Souza, Anna Maria Alves de) (My Science book)
Meu livro de Folclore. (Azevedo, Ricardo) (My Folklore book)
Meu livro de Pesquisas (Guimarães, Décio Gonçalves Ribeiro) (My Research book)
Meu livro de saúde. (Lamare, Dr. Rinaldo de) (My Health book)
Meu mundo Adolecente (Zezinho, Pe. S.C.J.) (My teenage world)
Meu primeiro dicionário. (Tersariol, Solange Feletto) (My first dictionary)
Meu primeiro namorado (West, Callie) (My first boyfriend)
Meu suposto namorado (Winfrey, Elizabeth) (My supposed boyfriend)
Meus três namorados (Page, Alexis) (My three boyfriends)
Minha escola e meu bairro (André, Julierme) (My school and my neighbourhood)
Minha irmãzinha é um monstro (Brezina, Thomas) (My little sister is a monster)
Minha Primeira Paixão (Bandeira, Pedro) (My first passion)
Minha profissão é andar (Pecci, João Carlos) (My business is to walk)
Minha vida de menina (Morley, Helena) (My life as a girl)
Some of those are even school text books, so these are not cases of slang or informal use of language.
Of course, it would have been perfectly possible for those titles to be accompanied by the appropriate definite article, giving:
O meu amigo canguru.
O meu filho tem medo.
The article can be used in most contexts:
A minha casa fica lá detrás do mundo,
aonde eu vou em um segundo
quando começo a pensar. (Lupicínio Rodrigues, Felicidade) (My home is there, behind the world,/ where I go in a second/ when I start to think.)
There are some contexts in which the article cannot be used, though. One that is quite obvious is in vocatives:
Você chegou atrasado, meu velho.
*Você chegou atrasado, o meu velho.
Not a complete answer, but to the question on variations in different standards and dialects:
The Portuguese version says that both standards allow both variants, but that there are perceptions that the definite article is used less in Brasil.