Could sina have been used here instead?
No, sina only refers back to the subject and is only used with 3rd person (han, hon, de).
Why not vårt???
Because "children" are plural. "Vårt" is for singular t-gender words. "Våra" is for plural.
Why "våra" and not "vår" ?
Våra is plural, vår is singular; children is plural
Would "We love our child" be the same too?
No, then it would be "Vi älskar vårt barn". Notice the difference on the possessive pronoun!
So when using sin,sitt and sinna we should look at the object if it is common, neuter or plural, why when using hans and hennes we take care about the subject's sexual gender?
Well, that's just how Swedish works, I'm afraid.