I am no expert but "min" is used for words that take an "en" article for example "en hund - min hund""a dog - my dog". Mitt is used for words that take the "ett" article for example "ett hus - mitt hus""a house - my house". Mina is used for plural words quite simply so, my dogs would be "mina hundar". There is no reason words take different articles, its just something that exists in many languages, its just the way they are.

Mina, dina, etc. are for plural. The rest as you said is used based on the gender of the word, for which you mostly need to remember what gender is what word, although living being mostly (not always) get en. There is also some criteria to guess the gender based on the letters that form the word but I can never remember it, sorry.