My live-in partner went home with me to Denmark for Christmas. The border crossing in Copenhagen Airport was pleasant and completely without incident.
What was maddening was the process of applying for a visitor's visa. In addition to forms, it required around 25 pages of auxiliary documentation, including her most recent pay-slip; a print-out of her bank balance on the day of submission; a letter addressed to the embassy from her employer attesting to her position, her monthly salary and her length of employment; a travel insurance policy covering the length of the stay; a round-trip plane ticket; a letter of several pages from me, accompanied by evidence such as photos, describing how and when we first met, when I most recently met her, the total amount of time we have spent together face to face, etc.
That was for a two-week stay over the holidays. Fun.
They are extremely paranoid of anyone "sneaking" in, especially from countries that could be considered "poorer". I agree, the process of a visitor's visa is insane. You are probably much better off if she just came as a tourist...
Also, when you are inviting someone into "Schengen"/EU, you are taking quite a huge responsibility because any and all costs for the state or hospital bills can ultimately be billed to you. So trying to see the upside of things, you could say they are just making sure all bases are covered.
On the upside, you won't get shot at the border - I'm looking at you, USA/Mexican border.
I'm UK/EU, but I've travelled around the EU with both my wife, and with friends, all USian (and all white, which I presume is still relevant in some places). The consensus has always been that the most difficult parts of their trip is getting back into the US.
We seem to have a 50/50 rate of whether they'll actually open your passport or not. Higher chances when flying in from the UK/US, lower chances when you're flying in/out of a city which isn't in the 'top 3' for each country.
The only question my wife has ever been asked, was leaving Italy; the chap wanted to know why she didn't have a stamp showing her date of entry. We had flown in from Zaragoza, Spain, and simply hadn't met any passport control. Deplane, baggage, and out the front door. We showed him a ticket stub showing that we'd entered the country less than 2 weeks ago, and got waved away.
The only noticable difference we've had between her (american) passport and mine (british), is that it is (or was?) difficult, verging on impossible to do an online check-in with RYR.
(to the original post; I don't travel to the US. ever. Mostly because I've put my 'status' in a gray area by walking out on a residency permit. You think it's bad when you're routine. hah!)
If you have a U.S. passport at least, my experience is that bored immigration agents wave you through. I actually immigrated to Denmark on a 3-year temporary work permit, and showed up at the airport as a single male in my 20s with three giant bags, and nobody asked me a single thing about it. I had the work-permit documentation, but they didn't even want to see it; just waved me through.
Supposedly you get more attention if you're Arab-looking, though (I'm rather white), and there can be visa hassles if your passport isn't from a western country.
I am an Italian, living in Brazil, my girlfriend is Brazilian. The only minor hassle she had was once in Portugal, when the officer asked her three times in a row what she was planning to do in Europe (probably because, from his point of view, she was a single woman from Brazil in her early twenties). It was an exception, as in my experience Portuguese officers are in general extremely friendly and professional.
She has also crossed borders in Italy and France, and was treated very well each single time; I am sure that the fact that she speaks Italian fluently and has an advanced level of French was a positive factor, but nonetheless she was treated just like any EU-citizen there.
