Of course Naruto cares that's why he wears the Uzumaki symbol on his arm like the other clans wear their clan symbols even though the flat jacket bearing the symbol is gone.
Personal relationships develop over time and through mixing especially once he'd learn she's of uzumaki blood. Their connection would grow stronger, I don't see any problem with it tbh.
Boruto is a clone is he completely identical I wouldn't say so but that doesn't change the fact he's a clone. Looking similar to your parents I have no problem with if kids didn't I'd be worried. However he's pretty much Naruto with slightly different hair style, 1 less whisker pair.
Dude, the Uzumaki clan symbol is simply part of many Konoha outfits. It is even on the back of Konoha flak jackets. That is not because of clan affiliation. It is just because of Konoha's friendship with the Whirlpool village. Naruto has been wearing that symbol since long before he met Kushina, and I doubt that he even knew that there was an Uzumaki clan before meeting Kushina.
I doubt he would care that she is of Uzumaki blood. Naruto (like myself) seems to care more about personality and how you actually feel/interact with that person rather than blood ties. Naruto has had very little interaction with Karin, and even if Hinata had died in the war like you said, that still wouldn't give Naruto any sudden incentive to go start randomly hanging with Karin.
Boruto isn't a clone. Having several similar features with slight modification is common for offspring. What exactly are you expecting? Sometimes kids come out looking way more like one parent than the other, and that was just clearly the case with Boruto. By your logic, isn't Sarada a clone too (black hair, black eyes)? This is common for kids.
