Translation:I have a dog in each one of my houses.
itastudent, as a native English speaker I would not use "one" after "each" in that sentence. I agree with AndreasWitnstein regarding "every"; "every one of my houses" would be correct, while "every of my houses" would not be correct. Incidentally, DL did accept the translation of "each" without "one" after it.
Yes, the ‘una’ is necessary. Omitting it makes it ungrammatical, but doesn't change the meaning —It would still be understood. If English is your native language, think of ‘cada’ as meaning “every”, rather than “each”: In English, it's also ungrammatical to say *“every of my houses” instead of “every one of my houses”.
You still have to worry about gender and plurality, and I don't always remember, but--and I guess it's a teeny bit of Latin I'm thinking of-- I think of the "est" in est(o/a/e)(s)(?) as indicating that the object in question IS right here and so is a this. But this may be too idiosyncratic a mnemonic to be generally helpful.
No, Tengo un perro en las dos de mis casas is not a correct sentence in Spanish either. In partitive sentences, we don't use ambas or las dos in Spanish; the same applies to las tres, las cuatro, etc. What you can say is Tengo un perro en las dos casas mías or Tengo un perro en ambas casas mías (all of them), that is different from Tengo un perro en dos casas mías or Tengo un perro en dos de mis casas (not all of them).