[...] started undergoing hormone therapy and estrogen replacements pre-puberty, [...]
There's no need for it to be pre-puberty. In fact, but I don't guaranty it, I more or less remember of a talk about this with a doctor, saying that it can in fact be a real problem to do the hormonal process before the end of the puberty. When a transgender (because your character is a transgender) pass through the hormonal therapy, (s)he live a second puberty. I assume that what the doctor was saying at this time was that it would have needed to double the dose, since at this time the organism was already boosting the hormonal production through the natural puberty process.
This said, for the surgery, it's not mandatory. Not all transgenders transition from their birth body to the body they should have had. And not all the ones that transition, go through the whole process. The main "visual factor" is in fact the boobs, that what people use to differential a girl and a boy, and so the body part that is the most usually removed/added (depending in the case, in yours it's "added", but not necessarily by surgery, the hormones can do a great job by themselves).
So, you can simply say that your character don't felt ready yet to be ride of his penis, or just ceased to feel a body dysphoria once she grew boobs. It happen.
But, there's a thing that happen in the majority of the case (but not always), it's the need to remove the sexual organs by themselves. I have few return coming from MtF (Male to Female) transgenders, but I heard way too many FtM ones talk about how it was really painful to keep their ovaries once the hormonal process really started to act. To
over simplify it, ovaries and a really high level of testosterone aren't really compatible, and it's like the body tried to destroy this part that shouldn't exist. So, your character will probably want to get ride of her testicles, or discover what women live once by month, but for her it will be more often than this.
But, again, this is also not something mandatory. It seem to happen in the majority of the case, but not always, and some transgender live it better that other ; probably because the level of pain isn't itself a constancy and it's more bearable for some than for others.
Are they male, or are they female or transgender female, shemale or futa?
What you describe is a MtF transgender. But you can also explain the absence of body dysphoria by the fact that she discovered "dickgirls" by reading hentai, and felt that it is what she really is ; neither the boy she was born as, nor the girl she firstly thought she effectively was, but really a girl with a dick. So, while being technically a transgender, she can identify herself as dickgirl without problem.
I'm guessing fucking them in the arse would come under gay,
Not necessarily. She's a girl, and there's absolutely nothing gay for a girl to take it in the ass. It's her partner that need to be bisexual or pansexual (a gay wouldn't be appealed by the boobs and the girl looking body ; and will not support long enough the girl personality), for her it's a straight sexuality.
But if you plan to make her have sex also with girls, then she must be bisexual, but will probably define herself as pansexual. In fact, it can be this pansexuality that lead her to the dickgirl option. She always felt appealed by both male and female, and so felt it more natural to stay this way.
It's honestly one of the most confusing topics I've come across
Hope it helped you.