How much detail to include in first email to potential PhD supervisor?
My answer is going to extend up this earlier answer to a similar, although broader, question.
Some background: I have worked with two pretty well-known professors in Austria and Switzerland, and can provide some insight into how they tend to hire. I assume other professors have similar MOs, but not every person is the same, so your milage may vary.
Indeed, for both of them, the first step towards starting a PhD is to send them a short informal mail stating your interest in joining their group. Your challenge is to get the professor interested despite him reading maybe a 100 mails a day. You can already see why a two-page text has a 0% chance of being read, same is true for an attached CV or an elaborate research proposal - you need to convince in maybe 10 seconds. Your second challenge is to separate yourself from the dozen or so other people that are trying the same every week, mainly coming from universities in the far east.
In that light, here are answers to your questions
Generally, how much and deep details should I provide?
I would go for none, honestly. Discuss concrete research ideas at a later point, when the professor has shown interest.
Should I mention anything about fund chances?
God no.
Should I send documents or only if s/he asks for.
Yes.
Statement of purpose/motivation, what is the right length? Is s/he ready to read two pages about every applicant sends him/her email?
Forget it. None of the professors that I worked with is interested in those formal application documents at any point during the process. They will either want to skype with you or have one of their senior staff skype with you, and then you would explain the things you would write into a SOP. To be clear - I have received a PhD student, a postdoc, and a senior postdoc position without ever being asked for a SOP or letter of motivation. Those are required only if hiring is done by a committee, e.g., for faculty staff.
Here is an example of a mail that might work on the people that I know.
Dear FIRST-NAME,
I am currently looking into options for getting a PhD. I have looked into your work on IMPORTANT-THING-PROF-IS-WORKING-ON, and I would be really interested in joining you on this line of work.
I have recently graduated from XY with a degree in YZ, and I have a background in SOMETHING-RELATED-TO-THE-ABOVE. I have done internships at BIG-NAME-A and BIG-NAME-B and already published X papers on SOMETHING-RELATED-TO-THE-ABOVE during my masters.
I would be happy if we could discuss matters further via Skype.
thank you in advance, YOUR-FIRST-NAME
YOUR-WEB-PAGE
Note the informal tone. This might be a personal preference of the people I know, but an informal mail is significantly more likely to be read by both of my professors than a very formal one. Further, note that it will be required that you have some measure of achievements that the professor can relate to - graduating from a university that the professor maybe has never heard of alone will not be enough to get him interested. Already having published and/or having done internships at well-known companies (well-known also to the professor!), especially those that are known to have competitive selection schemes (e.g., IBM), helps a lot here. Do not bother sending your GPA etc. - people in Europe tend to not be interested in grades in my experience.
Another thing that might help with the people I know is work on open source projects (e.g., being a committer or committee member to one or more Apache projects, having a well-maintained Github page with interesting tools and Gists, etc.), but this may be mostly because I work in software engineering (and people that know how software is built in real-life are very valuable to us).
Another important topic is english language - if you cannot speak / write english well, let the mail be proof-read by somebody who can (according to the way the question is written, this should not be an issue for the OP, but this may be relevant to other readers). Mails in terrible english are almost always discarded immediately.
Finally, the above sample contains a link to a web page. Have a personal academic web page. This is the place where you would put a good picture of you and all your academic achievements (papers, CVs, research interests, internships, links to open source projects you contributed to, industry projects that you worked on, whatever). Make it look professional and pretty.
Note that this mail still has a high chance of being ignored. In that case, give the professor a week or so and then write a a one-line reminder as a reply or forward of the original mail. If he does not respond after that, move on. Then he is just not that into you.
At most write an email about 6-8 lines long; people are busy. You should introduce yourself, explain what kind of research you want to do, ask he/she if she is taking on a student, and reference a few of their recent papers that you are interested and why. Include a small CV attached in the email.
If the professor responds, and offers to answer questions, then ask about funding. Statement of purpose length should be specified in the application process. Don't send your statement of purpose unless they specifically ask for it on their website.
It really depends on the field, but using these general guidelines almost every potential advisor responded this application season. Just because they respond doesn't mean you will get in, but it should improve your chances if you come off in a positive light.
Good luck.
I spoke to many supervisors in my applications for Computational Biology PhD's in the UK and got very enthusiastic and positive responses and lots of offers. They pretty all went something along the lines of:
Dear X,
I am a X student at X and I've been looking at your work in X and I'm very interested in doing a PhD in this area, do you have any positions available? Also do you have any time to Skype so we can talk about your work? I have done X which is why I am interested in X and my experience is detailed further in my CV (attached).
Best wishes, X
Obviously omitting/changing details if they have advertisements for positions out etc which you can then say you have seen, I recommend making your CV as strong as possible and also offering to Skype shows you are willing to spend time talking to them! I also sometimes asked them whether they would consider me or checked whether they felt I was complementary to the group/had the right skills.