Hiragana and katakana are both the same phonetically for different purposes; hiragana covers natural Japanese words as Katakana is for writing adopted foreign words. Therefore it wouldn't make sense to use them both in the same word but a sentence as a whole can encompass both forms of writing. (The only time I think it comes into play is when you're doing a name like スレビさん　but I would love to get a second opinion on this case for you)

I agree with what Toby san has answered. For example, we hardly write Philippines and the United States as ふぃりぴん, あめりか. I'd like to add another examples which we sometimes use katakana. We sometimes use katakata, when we write following words. キラキラ、テカテカ、(These two words mean "shiny") カンカン(the sound of railway crossing: ふみきり) and so on. These are just examples, and we also write these in hiragana.

Good question. But, it depends what you meant by "mix". I mean, almost all the time foreign names (nouns) are written in Katakanas; in this case writing Katakana and Hiragana together is fine. Sometimes, adverbs such as 'bikkuri', 'hirahira', 'kirakira' etc are written using Katakanas. And sometimes, normal words are also written in Katakanas. For example, I am listening to 2 songs now by Ikimono Gakari. The first one called "Natsu Koi" which in Japanese is written like this on the album "夏・コイ". And the other one is called "Koisuru Otome" which is written like this "コイスルオトメ". (By the way, the song "Koisuru Otome" is really beautiful so please try listening to it if you haven't heard it yet...) I hope my answer was helpful. Please ask if there are any confusions...^_^

In general, you don't want to mix them up within a word. However, they are naturally mixed together in Japanese.