Does this explicitly mean I have a snake? Or could a possible translation be "There is a snake at my house"?
Seems that is the case in both languages: "house (building), home, family, household." https://en.wiktionary.org/wiki/%D0%B4%D0%BE%D0%BC#Russian
У меня есть змея дома is not the most natural way to form a sentence like this. You would be understood and the meaning would be the same, but you would also be putting a slight emphasis on the fact that the snake is at home and not somewhere else.
Personally, I'd say у меня дома есть... 100% of the time -- native speaker here.
Okay i get why this is confusing me lol. I keep thinking in terms of transliteration, i.e. "I have at my house there is a snake", when this sentence is closer to "There is a snake at my house."
Or does У меня not apply to дома but rather дома's just thrown in the middle for no reason?
Why not "У меня змея дома"?
Literally it's something like "with me at home there is a snake" - i.e. I have a snake at home.
У меня змея дома - sticking my neck out as usual with word order questions, but to me this sounds like telling me where your snake is, rather than telling me what you have at home.