The idea usually was to accurately reproduce the text in front of them (which almost certainly wasn't the original), but all kinds of things could and did go wrong. Just as an experiment, try copying out something in whatever your native language is by hand and see if, working in a language you operate in on a daily basis, you can produce an error-free copy. The copyists of our texts had varying abilities with the languages and it shows in things like misspelling words several different ways and such.

Then there are reading errors. I don't know how much handwriting you have to read anymore, but again, making out someone else's handwriting can be a huge pain in the ass. We also have a tendency to want to fill in things we don't understand by inference from the things we do, so if a scribe misreads a word, he might try to construe the rest of the sentence in a way to make his original mistake make sense.

Then there are things like skipping lines, or repeating a line already copied, all the same kinds of reading mistakes we make every day. Pages get stuck together. Someone takes a break and loses their place. All of that will show up in the new manuscript.

Sometimes scribes realized there was a mistake in the text and tried to correct it, either by erasing bits, or writing in the "correct" reading (corrections aren't always right), or by sticking stuff in the margin. But they might also use the margin to include other info - a sort of cross reference. Now, a third scribe gets his hands on a manuscript with notes in the margin and has to decide if they are corrections that belong in the text, or if it's the cross reference sort of note, and whether it should be reproduced in the text itself or the margin. Or at all.

Modern editors of these texts try to sort out these errors and produce something like the earliest possible text we can - this is the field of textual criticism. We are pretty good at it (we can sometimes test our corrections against papyri, for instance), and in case we get things wrong, a good scholarly edition will have what's called an apparatus at the bottom of the page - a bit of notes that give important possible alternative readings. These are extremely brief notes in a sort of code; they are not for everyone to use. They scare me a bit. But they are there so a reader, if one wants, can evaluate the decisions the editor made in producing the text he or she produced.

There are places where some people suspect the text has been changed on purpose to suit an agenda. Bart Ehrman's book Misquoting Jesus looks at several of these kinds of changes. It is also a very accessible introduction to textual criticism generally.

The answer would vary greatly depending on the text in question. The extent to which we can determine what the original was (and thus what are interpolations) is itself highly dependent on the manuscript tradition and the number of extant witnesses (i.e. surviving copies). Yes, all the things you mention happen in places. Sometimes they are easy to pick out; certain interpolations or omissions (particularly omissions) have surely escaped modern notice.

To approach a manuscript tradition, you must first locate as many copies of the text as you can. Manuscripts can be dated, sometimes with astounding precision, based on their script and codicological features. This discipline, paleography, had its first stirrings in the Humanist movement of the Renaissance, but it has its true origins in the seventeenth century, when various monasteries attempted to assert the legitimacy of their privileges, often attested on manuscript charters, or "diplomas".

A handwritten copy will never be 100% accurate. Scribes skip lines, make spelling errors, transpose words, etc. By looking at these variations, you can create a "family tree" of the extant copies, though if you do not actually have the autograph manuscript (exceedingly rare), you can only approximate the original text. The art of reconstructing this stemma codicum (family tree of books) was actually first used by 19th c. German philologists to attempt to restore the "original" text of the Bible, but became generalized with its use in the massive (and continuing) Monumenta Germaniae Historica.

Scholars now use and create "critical editions" of texts. These have the author's interpretation (and without the autograph it is always an interpretation) of the original text, but provide footnotes with the variations found in each manuscript. These are invaluable resources for modern historians. If the text I am reading quotes a variation of another text only found in one manuscript of that text, that tells me something about the author!