You should not have just one CV, but you should tailor your CV based on the recipient. The distinction Veblen makes in his answer between resume and vitae is useful to bear in mind (although I do not make this distinction in what follows), and it is also important to bear in mind that expectations about what should appear in a CV vary between countries and industries. For example, CVs in Germany tend to be very long (e.g., I have just edited a German CV that is nine dense pages long), exhaustively documenting every post held, with every committee you served on at each post, every professional society attended, every journal for whom you have refereed, etc; and furthermore there is a strong expectation in Germany that the CV contains only objective information. In the UK, by contrast, CVs are expected to be short, most typically two pages, and it is quite acceptable to list only your most significant places of employment, and to add subjective information, such as what you consider to be your biggest achievement during the period you held a post.
In general, testimonials may be valuable in some applications, but they should be attributed, it should be clear why the testimonial is credible, and they are probably better in your cover letter than a CV, and if you do put them in your CV, I recommend that you have a testimonial section in your CV. They are more acceptable in the US than in Europe.
You ask about occasions where I'm only asked to send my CV and not a cover letter with it - this is a place where putting more and more subjective information into a CV may be useful. It is common for recruiters to want just your CV: be aware that the standard of ethics in the recruiting business is not high and you should not be too dependent on their services.
If in doubt, contact the human resources department of the institution to whom you apply before sending anything. Doing so demonstrates initiative, often will yield useful tips on an unofficial basis, might give you insight into what and how many applications there are for the post, and may help you avoid what the intitution regards as mistakes in a CV.
It is possible to provide supplementary information about your career if you have a page on a website, and provide the URL to this page at the top of your CV. It's common to link to Linked In pages, although bespoke pages offer more flexibility for you to tailor your presentation, include more subjective information, and probably will have higher information density and attractiveness than these networking websites can offer. I do not recommend putting exhaustive information about your career online: this material can be abused.