For instance there is no male or female, just one type. I doesn't have to be a mammal, but is there any organism apart from bacteria and the like that don't have genders?

Gender in animals is defined by chromosomes, so even the species which only have one 'type' still have a gender - there are many species which do this, including bdelloid rotifers and several parasitic wasps, including Dinocampus coccinellae, one of my study species.

Hmm, gender is an interesting one, there are species which are hermaphrodites, so they are both sexes (many slugs and snails for example) and there are plenty of species that can change sex from male to female or vice versa - sometimes they can also revert to their previous sex (clownfish are an example). In these cases the chromosomes don't change, so there is more to gender determination than just the chromosomes. There are also situations where a species has no males, such as in the New Mexico Whiptail lizard. In reality the lizards are all females, so they have a gender.

And as a side note, gender as a term can refer to things like sex, but it can also connotate a social aspect which isn't really applicable to animals (that we know of). If you're asking specifically at chromosomes and anatomy, it's more clear to just use the word "sex."

And just for fun, there are hermaphrodite slugs that will fight be tearing off each others penises forcing the losing slug to live out the rest of their life as a female!