... 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