What a wild ride you've been on with all this new news!
I can understand why you're hurt by your brother not mentioning this half-sister to you. I don't know him, but I can't believe it slipped his mind. I think he didn't want to get your riled up, upset, or something like that.
I'm the kind of person who's always interested in meeting new relatives, and researching genealogy. I think I'm just a curious person. Or maybe it's that the relatives I do know and grew up with are very toxic and so I like to imagine that somewhere, in my ancestry, or distant cousins, there are nice, normal people.
For this reason, in your position, I'd want to contact the half-sister, too. If she turns out to be a kindred spirit, you could have a new friend or acquaintance. If she ignores you or avoids you, you would feel rejected, but you would not have lost anything because you have no relationship now, to lose. You just have to go into it with no expectations and just be very careful.
My father has never told me that he might have a half-sibling, in a very similar situation to yours. However, my mother told me years ago that someone (a woman, I think) who was born around the time my paternal grandparents got married, contacted my father and his siblings when they were adults and said she was a half sister. So, it'd be like your half-sister contacting you. In her case, she never knew their father (my grandfather) and she just wanted to learn more about him, his family, his lineage, ancestry, etc. According to my mother, my father and his siblings were so disturbed by the fact that they might have a half-sister (meaning that their father either cheated on my grandmother or had an out-of-wedlock child before they were dating, and didn't tell her). My mother said that they (mainly one of my dad's sisters) shooed this woman off and told her that she couldn't possibly be their half-sister and that they didn't want to hear from her again or hear about their mother. And that's that.
Meanwhile, we all live in the (large) city where my grandfather's family has been since 1900, and I once in a while wonder if I have cousins out there who I don't know about. My generation is far enough removed from the drama that I would be interested and curious to meet cousins. And I wouldn't mind answering questions about the family history and genealogy that I know.
I say, mull it over for a few days, no rush. And keep any initial contact short, but warm and polite. And honest, if you would like to not involve your father. You should probably say something to the effect of what you said here: "For various reasons, I'm not in touch with our father. That's why I didn't know about having a half-sister. But once I learned about you, I was interested in contacting you, just to learn more about you" or something like that.
Email is probably safer, if you don't want her to pass on your last name to your father. Although, I agree with blarg314, that if you're too secretive, she's going to think it's a hoax, trying to get her credit card number or something. I think it is better if you do contact her directly, and not through a third party. I always like to "go straight to the source" and not have people speak on my behalf. Like the game "Operator," words and meanings get garbled the more they are retold.
If you do go with email, you can set your "From" name to your maiden name and leave your new last name out completely. I have a lot of friends who, even after they changed their last names, still have email addresses from when they were single and still have their maiden names listed.