First of all, I would like to recommend reading a post by Grammar Girl, The Write Practice, and of course Wikipedia.
I've seen many styles of this ellipsis usage. The more technically correct term for this "three dots" is points of ellipsis, where ellipsis actually refers to the omission, not the punctuation mark.
Most styles seem to agree that we should that Unicode character U+2026 for a points of ellipsis in documents. But when we type on the web, we usually type the dot character (some will call it a period). Some styles suggest typing dots with space in between, like this ". . .", others suggest that you don't need those spaces, that is you can simply type it like this "...". But one thing is certain, we always use exactly three dots for this punctuation mark.
Here is a good rule of thumb that all styles I've checked seem to agree: treat this points of ellipsis punctuation mark as if it were words, whatever words we left out, that is.
Another usage is to use the points of ellipsis for a slight pause in speech, which becomes widely used on the web, e.g.
I want to tell you something...
But you already know what I am thinking...
Well, never mind that...
Let's go there...
I will buy you a drink
which, to some people, can be quite annoying.
A points of ellipsis at the end of a sentence
When a points of ellipse is used in writing, it can be confusing if we want to use this punctuation mark at the end of a sentence, or between sentences. I believe that some style guides will suggest you to avoid this. However, I usually see respectable books use it like this,
Firstly, . . . Secondly, . . . Thirdly, . . .
Note that there are three sentences, not just one.
But that is the style that puts spaces inside the points of ellipse mark. If we don't put spaces inside, it would become: Firstly, ... Secondly, ... Thirdly, ...
According to the mentioned Wikipedia page, one style says,
There are two commonly used methods of using ellipses: one uses three dots for any omission, while the second one makes a distinction between omissions within a sentence (using three dots: . . .) and omissions between sentences (using a period and a space followed by three dots: . ...). An ellipsis at the end of a sentence with a sentence following should be preceded by a period (for a total of four dots).
I found one of this four dots used in one of the references I mentioned,
"I can’t believe that I managed to escape with my life. ... We’re lucky to be alive."
Let's get back to your questions...
Which one is correct?
The latest research says that the number of mobile Internet users will increase by three times in 2015. Read more... (Note the three dots after the word 'more'.)
The latest research says that the number of mobile Internet users will increase by three times in 2015. Read more.... (Note the four dots after the word 'more'.)
I would say it depends. But in this case, "Read more...." seems odd since there is no sentence after it. This is quite similar to the title of one reference I listed, How To Use an Ellipsis… Correctly.
1) Under which circumstances can we put more than one dot ('.') within a sentence?
2) Under which circumstances can we put more than one dot ('.') at the end of the sentence?
3) Is there any rule that talks about the number of dots to be put?
These questions were already answered in the beginning of my post. To sum it up, we always use three dots for a points of ellipsis. We usually use this punctuation mark to represent either omissions or pauses in speech and written text. You can find general rules in some references I mentioned, and for the exact rules you should consult your manual of style.
Hope this helps.