...␣part of a sentence␣...
...part of a sentence␣...
...␣part of a sentence...
Notice the spaces before/after the dots. Which usage is the correct one ?
This is a matter of pure style. I've worked in houses where the style sheet called for spaces before and after points of ellipsis, and in other shops where you close up the spaces fore and aft. What matters most is being consistent once you've selected one style or the other.
My preference is for the Chicago Manual of Style method, which closes up the spaces. There are other, more subtle rules about the use of points of ellipsis, and the section here in reference to Chicago explores some of the finer nuances.
One general rule to know, which is pertinent to your examples above, is that points of ellipsis are trailing punctuation - they follow words, but do not precede them. For example:
Right: "The archeologist opened the door of the tomb..." Wrong: "...opened the door."
Right: "He...opened the door."
You might start a line of text with points of ellipsis if you are writing creative dialogue in fiction, and are trying for some kind of special effect, but that is a matter outside the realm of formal composition.
When placing an ellipsis in a quote, it is like a comma, colon, semicolon, etc, no space before the "..." and yes space after.
EDIT: In chatting/texting lingo, it is common to indicate a pause before responding with a "..." without a trailing space
Choosing whether or not to include spaces between the ellipses and the words is mostly a stylistic choice, and often has to do with readability, such as whether or not the dot closest to the word tends to disappear into the letter next to it.
As for any meaning denoted by spaces and the lack thereof used in the same work, it is so varied in fictional works and formal works alike that it is a matter of internal consistency. When reading a particular book, a space before or after the ellipses may denote a longer pause or more complete thoughts, whereas the lack of a space may denote a more hurried and out-of-breath tone. When reading another book, the space and lack thereof may seem to denote the opposite. The only way to determine this objectively, in my opinion, is to take a line of dialogue that includes one or more ellipses that makes far more sense when taken one way than when taken the other way, and refer to that when deciding what the styles on the rest of the ellipses denote. I have yet to associate changes in spacing with anything other than changes in tone or pacing.
As for ellipses occurring before a line of text, this occurs often in graphic fiction, but almost always follows a bubble which ended with ellipses. This is there due to space constraints, and the inability to put a complete thought in one bubble. Less commonly, but in more mediums, this can indicate that someone is refusing to be interrupted and continues talking over someone else.