male and female possession
If I am a guy will I always use the masculine from for possession.
For example if my cat is a female, do I say: ¨La gata es mío.¨ or does should I then use mía for mine, because I´m talking about a female cat?
In the same sense if: They (the girls) read our (boys) letters. In my mind it should be ¨Ellas leen nuestros cartas¨ but the it was marked wrong in a test.
Is that a mistake in the lessons or am I missing something
The lessons are right. Every pronoun is supposed to agree in gender and number with the noun it modifies. Therefore, "la gata es mía" / "los gatos son míos" / "el gato es mío," and "ellos/ellas leen nuestras cartas." Consider this "el hombre tiene las manos sucias."
To clarify, in the case of "la gata es mía" etc. mía is not a pronoun but an adjective. The rules are still the same, match the adjective with the gender of the noun.
And, as Raymond said, when using a pronoun you still have to use the proper gender associated with the noun.
Although, something you may be interested in knowing is that the past participle of a verb, so long as it is not irregular, when referring to yourself will always match your own gender. "estoy ocupado" I am busy (and also it is inferred that I am male).