"Ho un serpente nello stivale."
"Ho un serpente nello stivale" - I have a snake in the boot
"Ho un serpente nel mio stivale" - I have a snake in my boot
That's the literal translation for them. In practice the first one could be used for "I have a snake in my boot" too - because it would generally be assumed that you are talking about your own boot.
I don't know which one is more common though.
That's what I would have guessed - it makes sense with how Italian generally works.
The Dover book sounds pretty good. Looks like it'd be just the thing to complement Duolingo. I'm not likely to buy a grammar book given I still have my old Italian textbook from uni, but others might appreciate your recommendation. :)