hey! i notice that your in seoul!
I am too!!!!
I waited for Coco to be 7 months to get her spayed. Because I didn't want to spay her too early. She just got spayed last week.
As for neutering, Mickey was 9 months when he got neutered.
And as for leaving ur dogs for 5 weeks they won't forget you

they know who their mommy/daddy are...
Will your dogs stay with the dog sitter? Or will the dog sitter drop by a few times a day?
Because they are still kind of young (also depending on when you guys leave)
I think maybe your dogs should stay with someone instead of someone dropping by.
Anyways, I'm sure you have a vet but if you don't, Coco goes to Itaewon Animal Clinic and Dr. Lee's her vet. Coco absolutely loves him! When she is being examined by him, she doesn't even know she's at the vet :P