Isn't cheating lying about it? Not simply sex or emotional intimacy with another
Some people need to form emotional bonds or do "sexual acts" as part of their employment. Actors and sex workers come to mind. Gynecologists, surgeons, and psychologists also may get into grey areas. To handle this flaw, it's common to define cheating as more to do with the lie rather than the act. "Yes my job involves touching other women's genitals, however it's not cheating, it's an important medical procedure.". His wife goes "Oh yes that's perfectly reasonable, I am informed about your occupation and consent to you performing your duties at work". Therefore zero cheating.
Only when he and one of his patients run off for a "private examination" in a hotel room does it become cheating because what is happening doesn't match what he described to his wife. He's lying to his wife, and thus now it's cheating.
However that safer definition is also an issue for your goal - notably open marriages and polyamory.
Me and my spouse have freely and with informed consent given each other permission to have sex or maintain secondary relationships outside the marriage with whoever we choose so long as we tell each other. She can sleep with whoever she wants with my consent, thus it's not cheating.
Also - if we invite a person to join us in the bedroom we're obviously not cheating, as we never left each other's presence.
How will your system deal with more complex rape cases?
Steve sneaks into woman's house, woman asks "John is that you?", Steve says "Yes, it's me, John". Woman and Steve then have hot, enjoyable, sex. Later the ruse is discovered and the woman is shocked, she'd never consent to sleep with Steve, only John. This is rape by deception, and results in consensual sex becoming rape after information is learned. What if this reveal occurs 20 years later? Or 50 years? Are they still put to death in the retirement home? Does the victims stigmata change after the ruse is revealed?
A similar situation occurs when defrauding a sex worker. Consent was given assuming conditions (payment), once the conditions are violated, the consent didn't exist. Therefore it was rape. I hope the goddess has a data feed to the bank to make sure cheques clear.
Brainwashing / grooming will result in sex being temporarily consensual, but then reclassified into rape once the brainwashing is broken. For example a cult leader. How will this be handled? "It was consensual at the time but now I think of it as rape".
Under (at least Australian, probably more) law when someone passes out they withdraw their consent, and it automatically becomes rape from that point. Marked for death is pretty black and white, and someone getting a bit woozy from over stimulation and starting to pass out in orgasmic bliss is a spectrum of grey.
People rape their married partner. Many societies struggle to recognise this a rape legally (Australian law for a long time defined rape as "man forces sex on a woman who wasn't his wife"). I define domestic violence rape as rape obviously, Does your goddess?
I hope no love potions are available for sale in your kingdom, as taking away someone's freewill so they fall in love with you, and thus have sex, is rape.
How would you handle mutual accusation cases? Two drunk people, neither of whom consented, both accusing the other of rape. Do you just kill them both? And if your policy is just "kill the guy" then what if they're both the same gender?
How will you deal with parasomnias? There are sleep disorders in the same family as sleep walking which can cause people to rape those nearby in their sleep, and then wake up with no knowledge of it.
How will you deal with child abuse survivors who imitate their attacks on those around them? This is a very real problem for children in emergency accommodation after being removed from abusive homes. I've done relevant volunteer work with abused children, and the most distressing thing I ever saw was a 6 year old girl who had been so badly raped, so often, that when given some time alone with a 5 year old girl attempted to rape them - it's how she understood human interaction. Is your goddess going to kill a 6 year old girl as a rapist?
Is IVF and surrogates cheating?
Scientists inject my sperm into my wife's eggs, which they then implant in a surrogate woman's womb. This is all consensual and paid for. How many of us are put to death?
Doesn't this system encourage divorce?
It seems once your divorced, you can live your life in peace without these rules. I see a festival in which 4 tennagers just getting into puberty, 2 guys and 2 girls (assuming heteronormative here) line up in a diamond, everyone having two people of the opposite sex next to you. The men get married to the woman on their left, then they turn to the woman on their right and make eye contact. While making eye contact everyone lists all the positive attributes the other, then talks seductively, then romantically, then dirtily, then explicitly if needed, until it classified as emotionally cheating, all 4 walk into the temple and are then "divorced", and are free of the curse of the goddesses smelly hand symbols forever, and they can have sex with whoever they want without being executed.
This awkward festival could be integrated into high school sex ed, with the trip to the temple being a sex-ed field trip educating them about the past beliefs of human sexuality.
You can get used to any bad smell eventually
I don't think I need to say too much about this, but if you entire society stinks of the same bad smell, within a few days no-one will be able to smell it, it'll just be what people smell like.
Within a few generations, someone whose remained a virgin until marriage will just smell wrong.