Portuguese is a pretty obvious mix of Spanish and French. I wasn't aware there were other predominant influences (though there always are)

And you're right about English, there's a joke about how our tongue stalks other unsuspecting languages in dark alleyways, knocking them out when opportune to steal loose grammar and vocabulary

What makes Portuguese a mix of Spanish and French and not, as TelmoMenezes says, yet another child of Latin? How do you categorize Romansh, Languedoc, Galician, Sardinian, and all the others?

Most of the words I've encountered in written Portuguese appeared to be either Spanish or French, or visually a combination of the two. It's another child of Latin, alongside Spanish, Italian, French, Romanian, and others. I don't know the history and I don't intend that Portuguese is a child of the two, only that it is another product of the Latin echo chamber by way of its two most visually apparent influences.

I've never had any purposeful encounters with any of the languages you mention, though I would certainly like to!

Is Brazilian Portuguese a hybrid? It borrows an awful lot from English, African and native American languages.

Not at all. It sounds different from European Portuguese, similarly to how American English sounds different from British English, but they are the same language, and mutually intelligible.

There are local dialects influenced by the languages you mention and others, but Portuguese is Portuguese, like English is English, with the normal variations you would expect from a language widely spoken around the world. European and Brazilian Portuguese native speakers can communicate with each other easily, share TV shows, etc.

Of course this is anecdotal, but my wife is Brazilian (from Minas Gerais) and struggles to talk with European Portuguese people (sorry!), and needs subtitles to understand it in a film.

Even in my Portuguese textbook it says that there is significantly more difference between European and Brazilian Portuguese than between UK and US English.

No need for apologies :)

I think the main issue you are alluding to is phonetic. European Portuguese speakers tend to really close their vowels, while Brazilian speakers open them. The end result is that it is easy for Europeans to understand Brazilians, but not necessarily the other way around.

I have seen this happen with English speakers, by the way. I remember a friend from the US needing subtitles to follow a British comedy show.

Would your wife have any trouble reading this, for example: http://publico.pt ?