Well, to be blunt, if you violate the Sabbath in some parts of the world (say, KJ), the only reason they WON'T execute you is because they don't have the authority to do so. If they could, I expect they probably would.
Israel is governed largely by secular Jews who don't even worry about touching across gender lines, let alone other violations of the Torah. So that's not relevant.
I would strongly advise you, if you are ever in KJ, or any place like it (there are several such communities in the USA, and quite a few in Israel), DON'T go and break the rules of Torah. You will be ridden out of town on a rail at the very least, and possibly worse. For example, show up to KJ some time, ladies, wearing daisey dukes and a halter-top and start wandering around. I can pretty much assure you that within about five or ten minutes, you WILL get stones thrown at you until you either one, leave, or two, are seriously injured or killed. And no, don't expect some "good Samaritan" to call the outside authorities to help you. Won't happen. The same is true for men who misbehave or get into mischief.
So, remember. Just because the Jews you SEE don't kill each other for violating Sabbath doesn't mean it couldn't happen. It just means that the Jews you see in the public eye are the liberal Jews or the non-religious Jews. The same Jews that touch each other outside the family are the ones who don't worry about violating the sabbath. The ones who won't touch each other are the same who would not violate the Sabbath, and so would do fine in some place like KJ.
And as men are to women, so women are to men. The restrictions are the same. Gay Jews are not contemplated in this scheme of things, obviously. When the text was written, no one really gave that much thought, I don't suppose. At the time, homosexuality was understood to be forbidden outright under penalty of death. As we've come to differently understand human sexualities, I am not sure what that may or may not do to the traditional standards. Probably nothing. Since the majority of men are straight, as are the majority of women, I expect that there won't be much change in the question of who may touch whom, especially since the Bible has spoken clearly on the matter. The Bible never spoke clearly on the matter of homosexuality. But on the matter of who can touch whom, it does.