My ds (13) tells me he hates me frequently, or that he thinks I hate him (normally because I've said no to something, or made him do his homework).
I have a fearsome temper, much better controlled now I'm older, wiser and learned to laugh at teenagers, as I teach them, and vowed never to be like my Dad...my line is that I really dislike your attitude/language/behaviour, but I love you, so you are distinguishing between the person and what you are seeing/hearing.
As for the 'dirty on the inside bit'; not perhaps the most tactful way of putting it, but for some people who are religious, or have a particular moral code, and the OP could be a Muslim or Hindu or evangelical Christian, for all we know; then yes, that is what they would think, even if it shouldn't be articulated.
I'm not sure an apology is the way to go - but a frank discussion with your daughter about what is going on in your family and in her life might be in order, with the promise of no shouting or recriminations on your side. When we have a discussion like that (not often thankfully, I revert to form tutor mode, and try to detach myself, as it is the only way I can listen without emotion clouding my judgement.
Don't beat yourself up - teens are hard work, and none of the come with an instruction manual, nor are we given training.
I find counting backwards from 10 in Flemish helpful, you could try it in French? By the time I get to nought, I've forgotten what I was wound up about.