How to approach American English vs. British English spelling in a paper?
I typically don't worry about this question as a writer. For what it's worth, I'm based in the United States, so I default to American English. I always spell check my papers, but in the more than 25 papers that I've submitted, I've never gotten any feedback about using American spelling rather than British spelling; this question is very relevant for me since many of my papers are on graph colo(u)ring. It may be useful to note that in the journals that I submit to and read, I've seen both American and British spelling. (I think many editors typically let the authors choose which they prefer.)
So my suggestion is this: choose either American or British spelling and stay consistent throughout your paper. Don't lose any sleep over this. It will not be the deciding factor in whether your paper is accepted.
However, for doing a literature review, I feel your pain...
Whatever style guide your use, citations should be cited exactly in the original spelling. Doing otherwise would defeat the whole purpose of citation: uniquely identifying an existing publication.
I could find no direct quote from APA to justify this (except that, well, they don't indicate that you have the liberty to edit the title of cited works!). However, multiple secondary sources make it clear, as for example:
Notes: Please "copy" the title of a book/an article/whatever (as far as the spelling of words such as "behavior"/"behavioral" are concerned) […] exactly as in the original.
Citation style guides tend to be more explicit for foreign language works, where most of them explicitly state (in some wording or another):
Give the original title, and, in brackets, the English translation
If your question were about the -our/-or distinction, or the -re/-er distinction, it would be a real question, but with -ise/-ize there's no contest: go with the Z form, as it's accepted on both sides of the Atlantic.
The International Organization for Standardization favours British spellings, and uses the -ize form. (This is known as Oxford Spelling.)