I'm a little confused how you seem to put objective and absolute morality into the same category, rule it out, and then say it's still relevant.
As for your other points, every moral decision is made on a case by case basis and the arguments for why it would be moral or immoral to perform action X, Y or Z and which is most moral is tied into exactly the moral standards implied by the wider social contract. In a society where sexual fidelity was not the norm and where sex is a recreational activity much like playing tennis or sharing a pizza and watching a movie then there would be no moral "crime" committed if someone slept with someone other than their romantic partner.
Taking the act of sexual infidelity in our current society (the western one anyway) it still depends a lot on the context. There are unmarried couples who have been together for a long time where the implied or explicit level of commitment would make any sexual infidelity very serious, but I also know people who don't think there are any rules about that in the first month of dating someone new, and it's only if it gets beyond that point that it becomes an issue. There are also, to take it the other way, married couples who by mutual agreement are free to have sex with whoever they want, as long as it's just sex, much like in my initial example of an alternate society.
On the example of executing criminals, then the issue is the harshness of the punishment being properly relative to the crime committed and the certainty of the prosecution being sound and there being no doubt of an incorrect conviction.
These are the kinds of moral balances that we all face every single day (politicians obviously more so on a theoretical level) and I would contend that throughout the world the societies where these kinds of discussions take place and debate is had about the merits of one moral judgement or another on any given action are genrally speaking the ones where we see more "justice" (assuming no systemic corruption), whereas those societies where religion is blindly used without any thought about "is this actually the right thing to do" are more susceptible to moral miscarriages because using a scriptural basis for morality means you tie yourself to the moral standards of the society when those scriptures were written, and not the moral standards of society today.
And this kind of brings it full circle back to the point, that a moral judgement of an action in 2013 america will often be different to a moral judgement of the same action in 2013 saudi arabia, and will even more often be different to a moral judgement of the same action in 500 BC palestine. Judging people by the standards of a social contract within which they do not live and are not reasonably expecting to be judged by is not fair or just, but this is exactly what is happening when religion gets used as an arbiter of a moral argument. It's not that we are more or less enlightened or whatever, it's simply that the rules change with the society, and judging anyone by the rules of another society, whether separated by geography or culture or time or all 3, is not particularly productive for maximising common good or common happiness.
