Can a Ph.D. student who only knows how to speak English study in a European non-English speaking country (e.g., Austria, Netherlands, Switzerland, Italy, etc.) without any problems? Does applying to these universities require knowledge of that country's language?
Yes, they can, at least in technical fields (I speak mainly for Italy, but I'm pretty sure that this is possible in other countries around Europe as well).
To give you an example, at my university (Politecnico di Torino, Torino, Italy), there are a lot of foreigner students enrolled in various kind of engineering PhD programs who don't speak, or barely speak, Italian. In fact, most of the graduate courses are delivered in English. However, I recommend learning Italian (or any other local language if in a different country) at a discrete level if one plans to further pursue here (or there) his or her career.
I can tell from my wife's experiences who, as a native bilingual speaker (Chinese-English), tried to be admitted to Swiss universities: It doesn't work. For bachelor's and master's courses the local language (German, French or Italian) is strictly required. There are a few exceptions, especially for natural sciences, finances and economics, etc. (e.g. ETH Zurich, EFPL Lausanne, HSG St. Gallen, et al.). But for most fields the local language is a requirement.
However, this is totally different at PhD or Post-doc level. There you only need to find a professor willing to supervise you (which means that you must be able to communicate with him...), then anything's possible.
Also note that the visa requirements for Swiss student visas sometimes explicitely state that the knowledge of a local language will be put to the test in an interview.
Also note that there are often single modules taught in English sometimes even on Bachelor/Master level but they are normally an exception, so knowledge in the university's local language is still required.
Further note that some Swiss universities even offer courses where explicitely TWO local languages (e.g. German and French) are required or at least expected. I personally have experienced that several times.
My experience was in France, Université Paris-Est, to be precise, but the same applies, at least, for ESIEE, Marne-la-vallee, Ponts-Paristech, obspm and the Institut de Physique du Globe de Paris. I had direct contact with people from these places.
Professors and students speak english when they have no other choice, usually a very accented english (They have real trouble pronouncing 'w'). Speaking to you, they will indeed try to speak english, but amongst them, with you 'nearby', they will, most likely, speak french.
Most of the other employees will not speak english. In my university, the "welcome" session, presenting the university and the academic requirements and procedures, was held in french, even after I reminded them that one of the new phd candidates did not speak french at all.
Same applies in all public offices (including immigration, tax services, etc). Don't expect any of them to speak english. You might find someone, but don't count on it.
Of course, worse yet for random people in the street. My own landlady didn't speak english... Funny enough, all the cleaning ladies I found were either portuguese or brazilian, so that was easier :)
IMHO, while it is indeed doable to get a PhD and live in france without speaking french (I know a guy who did, although I think he learned french by the end of it), it is considerably harder. The academic part is easier, you can always ask your advisor to intervene, but the rest is painfully complicated.
I am a Dutch PhD student, and I am doing my PhD in Barcelona, Catalonia, Spain. In The Netherlands knowing only English as a PhD student is no problem. In fact in my university in the Netherlands (University of Groningen) master classes where given in English if there was a student who did not understand Dutch well. Also in daily life English is sufficient, especially if you are in a city.
In Spain the knowledge of English is worse, but at an academic level you normally can communicate in English. Also at my university (Universitat Politècnica de Catalunya · BarcelonaTech) most classes are also given in English. Here however, especially in daily life, it is very useful to know the basics of the local language (Spanish/Catalan), also for talking with some of the supporting university staff for example.
Edit: Some extra resources:
In the Netherlands there are around 850 Masters programmes taught in English. You can search for degrees fully taught in English in our database of Dutch degrees. ...
Knowledge of English is so good that it is possible to survive almost entirely in the English language (and watch BBC television without paying the licence fee). British students who have gone to study in Holland recently have found it to be a very welcoming and supportive study environment.
It is essential that you speak, read and write English well. You must have passed an English language test. IELTS and TOEFL are commonly accepted, but institutions may accept other tests as well, like like Cambridge English.
The required scores are at least 550 (paper based) or 213 (computer based) for TOEFL. For IELTS a score of at least 6 is required.
You can find language requirements for individual programmes or courses in the database of international study programmes.
An interesting document about the Dutch education system, also indicating that English is getting used more and more:
The language of education is Dutch, but under the influence of the Bologna process more and more study programmes are being offered in English. Education is compulsory in the Netherlands between the ages of 5 and 16.
Finally, a website where you can search for studies in English in Europe: Study in Europe in English
I know PhD students in Germany and the Netherlands without knowledge in German and Dutch respectively. Most of the PhD programs in Germany do not require courses. At least in engineering, most universities allow an English thesis. But to live in a foreign country without proficient knowledge of the language can be hard. In my experience, most people in the Netherlands, Germany, and the Scandinavian countries can speak at least basic English. In France on the other hand it is problematic without knowledge of the French language.
When I had made an inquiry about 3 years ago, a b-level proficiency of German language was required for most undergrad and postgrad programs affiliated to German universities. However, I think for a PHD, most universities do not require this. Rather, they would probably ask for your TOEFL or ELTS score. However, some programs might still require a working knowledge of German. I found some details on this site: http://www.findaphd.com/study-abroad/europe/phd-study-in-germany.aspx.
Summing up the other answers and adding some comments from my own personal experience:
Writing and defending a PhD thesis in English is possible in all the countries you mentioned. In some countries (e.g. the Netherlands), it's the rule and even the local students do it in English. In others (e.g. France and Germany), writing your PhD in the local language is still the rule and you might get strange looks if you choose to do it in English as a local (depends a bit on the field and on the institution as well). But PhD candidates from abroad can definitely do it.
Teaching, attending courses and dealing with the bureaucracy will be more difficult, knowing the local language will make you much more useful and open up many options. That's true even in countries/institutions that are moving to English for master's level courses and even more so in those where teaching is almost exclusively in the local language.
Life outside work will differ a lot. In France or Germany, you are expected to speak the local language for everything, taxes, shopping, etc. Most people will not feel confident speaking English. The Netherlands is a bit more English-friendly, films are not dubbed, people will spontaneously speak English when they hear a foreign accent, you can get some help in English even for official things like taxes (although not everything will be available in English), etc.