I'm trying to work out the rules around using the word cuid. Is it the same as with counting, where you use the singular form of the noun in the genitive case?

Ó Dónaill gives the following examples:
1. Mo chuid éadaigh, bróg - my clothes, shoes
2. Do chuid leabhar, eolais - your books, knowledge
3. Ár g~ saothair, trioblóide - our labour, trouble

In example 1 éadaigh seems to be gen. sg. of éadach but bróg is clearly nom. sg.
In example 2 I'm assuming leabhar is nom. sg. and not gen. pl. but eolais has to be gen. sg.
In example 3 both are gen. sg.

So, can anybody please tell me what's the pattern here? How do I know which form to use? How do I express "my exams"?

Mo chuid scrúdú/scrúdaithe?

bróg and leabhar are genitive plural.
Both are countable.

éadaigh, eolais, saothair, trioblóide are genitive singular.
All of them are (usually) uncountable.

Exams (scrúdú, scrúduithe) are countable, so ...
... what do you think?

Mo chuid ...
1) ... scrúdaithe (gen. sing.)
or
2) ... scrúduithe (gen. plur.)

I think I'm suffering from choice paralysis.

Thanks very much for the information.

As a follow up, where can I read more about countable vs. uncountable nouns in Irish? What is it based on?

the same thing as in English...
Basically, in Irish, they are singular nouns (in form) but in the meaning, they represent a group of things or people, or a quantity of some liquid or matter or something more abstract (wood, cloth, sand, knowledge etc).

_________________
Is fearr Gaeilg na Gaeltaċta ná Gaeilg ar biṫ eile
Agus is í Gaeilg Ġaoṫ Doḃair is binne

That may be easier to internalize, when you figure out, that it translates pretty well literally to English and actually works the same:

my share of shoes (gen. pl.)
my share of books (gen. pl.)

my share of knowledge (gen. sg.)
my share of trouble (gen. sg.)
my share of sand (gen. sg.)
my share of water (gen. sg.)

clothes in English are plurale tantum – there is no modern singular form of that word with the same meaning, but you still can say my share of cloth or better my share of clothing.

Ah, right! Stupid me! I was conflating countability with counting in Irish where the noun is always singular, "bád amháin, dhá bhád, trí bhád...".

Thanks very much to both of you for the insight.