No, it is not an F. It's a sj-sound. If phonology isn't your thing and the sound example in the link doesn't do it for you, think of it as something like the German and Scottish ch-sound (in "ach" and "loch"), but lighter.
Exactly, sin always refers back to a subject in the same sentence. So pojken sjunger för sin katt means that 'the boy sings to his cat'. But if she sings to his cat, we'd say hon sjunger för hans katt.
Is singing for the cat the idiomatic way one would call for the cat in the US? Like call in from the back yard? Or is our girl just a whimsical soul who sings for her cat which then purrs and cuddles up in her lap in delight?