"Ho un serpente nello stivale"
I'm not an expert on Italian, but I know that in French and Spanish - two other Romance languages - you generally don't use possession to refer to clothing. Same with body parts. They tend to refer to them much more... absently than English-speakers do.
I'm also not an expert in Italian, but I do speak Hebrew fluently. In Hebrew, it's the same way. It's assumed you're referring to your own boot/body part unless you specify otherwise.
Even in English, we can say "I have a pain in the neck", but it does seem that there aren't many instances where you can get away without the possessive.
It's implied, I guess. You couldn't say (at least it doesn't sound right to me) "I have a snake in the boot" in English, but in Italian, I even think it's the natural thing to say. That's how it is in my language, which is Romanic also.
I think it would be "nel mio stivale" (not "nello mio stivale", as you wrote). Because you have to pay attention to what comes after the member (la, il, lo) or the preposition (like in). Just like it's l'amica, but "LA mia amica" (not l'mia amica).
Google agrees with you on this -- 32,400 hits for "nel mio stivale" -- people find a LOT of things in their boots!
In the past, they've allowed us to translate "I have" as "There is". Why isn't that the case here?
What would be the difference between this and 'there's a snake in the boot'?
There's no difference. But you would rather say "There's a snake in this/that boot" which is "C'é un serpente in questo/quello stivale".