The IELTS website has information about their tests. For the General Writing Test they say:
In Task 1, test takers are presented with a situation and are asked to write a letter requesting information or explaining the situation. The letter may be personal, semi-formal or formal in style.
In Task 2, test takers are asked to write an essay in response to a point of view, argument or problem.
General Training Writing: What is the IELTS Writing test?
They say the following about the Academic Writing test:
Responses to both tasks must be written in an academic or semi-formal/neutral style.
Academic Writing: What is the IELTS Writing test?
Contractions and abbreviations like the ones you list are not formal or academic, so you shouldn't use them in the Academic Writing test. You can (probably should) use informal language if you're writing a "personal" letter in the General Writing test (task 1). Task 2 of General Writing is supposed to be "slightly more personal in style than the Academic Writing Task 2 essay" according to an official teacher's guide.
All of the words you mentioned are informal. Contractions (including "they're" and "I'll") have rules to when you can use them. For example, you can't use them at the end of a sentence (see this question).
I consider "ex" to be very informal. As a matter of style, I would recommend avoiding using it. It doesn't make anyone's writing sound more natural, at least in my opinion.
"Y'all" is something only used in the southern US, so I would recommend against using it (unless you're from that area). It's also very informal.