I totally agree, the way Aristine react is in character but the way it's talked about by everyone else as unnatural and like you can't truly love multiple people at once is weird. Plus you add to this that Tarkan isn't in love but under a spell and it make all of it even worse.
I wasn't prepared for polyamory hatred, especially since the founder of Iguro had a whole harem ? What a weird choice to introduce a character of legend with multiple lovers and then go on with every character acting like it's abnormal.
No even if he didn't cheat he's still searching for her and like everyone in the whole palace knows who's his first love is so he might've acted like that which gave it away and when Rine talked to him about this he couldn't give her a definite answer like "I love you" Or "I'm just searching for her to thank her for saving me or smth not because I love her" Even saying those things might've helped the fl a lot I'm not saying it's weird to have a first love but dating someone else while remembering your first love is the same as cheating it might not be physical but still
No. Cheating is doing relationship stuff with someone else without your partner consent. Thinking of someone else while it can be rude is not cheating.
There's nothing wrong with wanting or needing a closed relationship, but what you are describing, considering feelings as cheating even tho the person didn't act on it, is just toxic. Feelings aren't controllable, you can control how you react to them but you can't control how you feel and love isn't something you have in finite quantity. Some people can deeply love multiple people, some can't, what's important is to respect how the partner you chose feel and not act on those feelings.

The second princess probably has charm powers or something that ensnares others hearts, or maybe some form of mind control, hypnotism, brainwashing. Like it seems extremely obvious to me that tarcus's obsession is not natural.
I bet also that either she was the one who put the curse on the older brother that prevents him from returning to the capital or disobeying orders, or her father has the same power.
Unrelated, but i honestly don't get what's the bid deal if tarcus has a childhood love. Like I get our MC is emotionally stunted and naive in some ways about the heart, so it makes sense she'd feel a childish sense of betrayal about it, but like. He never cheated on her, he's not planning on cheating on her in the future, and the so called 'other woman' is a memory he has of a little girl, from when he himself was a child. This drama is so blown out of proportion. I don't mean to be unsympathetic to our mc, i find this conflict to be very in character, i'm just a bit skeptical of the overall message of the story itself. I wish there wasn't such obsession with having virgin male leads who can't even be attracted to other women. It totally misunderstands and distorts the meaning of what actual commitment and fidelity is, which is about choice and integrity, not being literally neutered when it comes to other women or not having experience at all??? Lol im digressing but that's just my two cents.
Side note, I enjoy the depth and complexity of all the characters. Love our mc, especially her contrast to her previous life. The brother was an absolute icon even though he only got two seconds of screentime. I also like the 2nd prince's character. He was born with a kind nature, but got twisted by the world from his exposure to human nature. Honestly, he's very interesting. The king is a complex character, reminds me of the father blacksmith; both of them breaking out of the mold of stereotypical 'good' or 'bad' dads we see in manhwa a lot. The queen is also very cool, like there were a couple moments where she really stood out to me, you can tell she truly loves her son.