How does authorship work in large collaborations?
How is it determined whether someone has had a signifficant contribution to yield authorship?

Very large collaboration are common in physics. I guess it started with collider physics experiments, and then it became common elsewhere (see LIGO). The general rule in very large (hundreds to thousands of authors) collaborations is that everybody who passes a minimum threshold of contribution to the experiment is an author. 

More specifically, an author is typically somebody working in an institution who pays money/puts work into the building/running the experiment. He/she has to have served at least a number of months up to a year helping build or operate some of the experimental equipment before becoming an author. Some collaborations make this threshold a one-off, others a yearly commitment. 
Interestingly, building the necessary hardware is seen as a natural qualification for becoming an author, while building the software infrastructure does not always count as a qualification for becoming an author.

Exceptions can be made on a paper-by-paper base when somebody who is not a typical collaborator but has contributed significantly to a paper is added to the author list of that paper only.  This is the typical case for theoretician who would like to use the experiment's data to prove/disprove a specific scenario.

Junior collaborators become senior enough to join the author list, less junior collaborators leave the collaboration, new institututions join the collaboration bringing in new collaborators. As a result, no two papers from the same collaboration in principle have to have the same number of authors; tracking the list of authors becomes another task to add to the long-list of large collaboration administrative jobs. 

How are authors listed?

Given the impressive collaborators list length (ATLAS and CMS papers at CERN list approximately 2000 people each) then the simplest choice is to list all authors alphabetically. Such a large number of collaborators publish typically  of the order of 100 papers per year. It is thus natural to expect that no author will ever read all the papers produced by her/his collaboration. Realistically, in such instances the average authors read 15-30% of papers he/she signs as author.

How does one determine the major contributor?

There is typically no difference between a null/negligible contributor, a significant contributor, and a major contributor in the author list. The only exception to my knowledge of the author lists "rules" above is for the Belle collaboration. In Belle papers, the main authors are listed first, with the others following in alphabetical order. If a reader would like to contact a Belle paper authors, he/she can write them directly. In all other instances, a reader would write to some committee that might eventually redirect him/her to the actual authors. 

How is the paper drafted?

The main authors of the result draft the paper; this is typically a team of 3 to 7 people, although much larger teams are possible for very important papers. The paper is then heavily scrutinized by an internally assigned committee, then handed over to another committee, then handed over to the another committee; in some instances handed over a fourth committee. The typical time it takes to work on the paper is thus in the hundreds of days - typically as long or longer than producing the result itself. However, scientists typically start working on some other projects while waiting for the committees feedback.

Does everyone in the collaboration approve the paper?

Not literally everyone. After the committees work, the general rule is the "opt-in". However, just as any author is allowed to comment on a paper before it gets released, any author can decide to opt-out of being an author for that paper (although it happens only in exceptional and heavily controversial instances).