score:5
You have to make a distinction between the Schengen area on the one hand and the UK, Ireland, Cyprus, Romania, Bulgaria and Croatia on the other hand. All EU countries should grant your wife the right to travel together with you (but not alone) with minimal formalities but those that are not in the Schengen area might still require her to get a separate visa.
For the Schengen area, you only need one visa. So if you want to go to Switzerland (technically Switzerland is not in the EU but for this purpose, it's pretty much the same for the time being), you would apply to the Swiss consulate. With the visa issued by the Swiss consulate you will also be able to travel in most of the EU without any other formalities (but, again, not to the UK, Ireland, etc.)
If you really want to visit all non-Schengen EU countries, then yes, you might need up to 5 or 6 visas (although Romania, Bulgaria, Croatia and Cyprus have some special rules for Schengen visa holders so it might be possible to go there with your wife's Schengen visa as well).
Furthermore, being married should in principle make things easier (at least in the EU proper, not entirely sure about Switzerland, see comments). In your quote, “these” in “check what these are” should be something like a marriage certificate, not the whole documentation package (insurance & financials & itinerary) that other applicants have to provide and the consulates have very little discretion in deciding to issue the visa or not. The visa should also be free of charge (regular Schengen visa fees are currently EUR 60).
Beware, however, all those rules only apply if you are travelling together as they are derived from your right, as EU citizen, to travel freely in the EU. Note that, since you are a UK citizen, these rules also do not generally apply to the UK. Going there might even be more difficult, depending on UK law.
Upvote:2
To answer the main question first, yes you need a visa. The situation is as follows. If you and your wife would life in a EU country other than your home country (UK) and she has a resident permit there you can travel to other countries without a visa. E.g. You both live in Germany and want to visit Paris. If she has a German resident card and you travel together she needs no new visa. Since you live in China (outside EU) you always need a visa.
Now about the insurance thing. As a spouse of a EU/EEA member she does not need to show proof of financial means including health insurance cover. You might still want to make a travel insurance for other reasons. You also don't need provide accomodation information.
P.S. as a general advice you might want to take an English translation of your marriage certificate, because when you arrive in Switzerland it makes things A LOT easier if you can prove you are a couple on holiday. (Single Chinese woman -> 1000 questions, with EU husband 2 or 3 questions)
P.P.S You need to apply for only one visa in the Schengen area where you arrive first and spend most time (transit does not count). In your case it would be Switzerland.