A chairde!
I'm about to tear "mo chuid gruaige as mo cheann"!
I learned that lenition occurred in singular masculine
in the genitive case
after the article, with the proviso that t and d
did not lenite and that s lenited to ts. Also that feminine singular nouns in the nominative case after the
article lenited with the same peculiarities.
Here's where it gets crazy for me. I also learned
that masculine and feminine singular nouns both lenite
as above if they follow the prepositions de, do or i
plus the article(den, don, sa)and that they lenite
as above, i.e. s goes to ts, t and d are unchanged.
I also learned that the other prepositions (except
gan, idir, &c.) plus the article eclipse the singular
noun, masc. or fem.
But I saw a website that gave "don sagart" and
"sa samhradh" along "sa tseachtain" and "don tsla/inte".
Is it that don and sa do NOT lenite following masculine
nouns or just do not lenite following masc. nouns beginning
with s?
Even more mind-boggling...That website gave
"as an tsli/" and "ar an tsra/id" alongside
"ar an salachar" and "leis an saol". Is there some rule
that feminine nouns beginning with s "eclipse" to ts
after article plus preposition. I never heard of the letter s mutating in situations where other letters
eclipse. Thank you very much!

Feminine nouns beginning with an 's' prefix a 't' after the article, even if it is combined with a preposition. 'Sa' ('in the') has an invisible 'n': insan. Therefore, ar an tsráid, sa tsráid. With masculine nounse 'leniting' an 's' after the article by prefixing a 't' is optional, the choice of whether or not to do it determined primarily by dialect. 'Sa tsiopa' is usual in Ulster and Munster, 'sa siopa' in Connaught (and in the Standard, even though the underlying rule for that does not apply there).

A Sheosaimh, a chara,
Go raibh maith agat...for your help. I have another
question for you to answer if you would...
Which is correct, or if both are, where are they correct
of the following? "ag an doras" or "ag an ndoras"?
Would the gender of the noun affect the operation of the
rule? And is it true that the letters t an d never
lenite after an, don, den or sa? Thank you.

Tá an dá cheann ceart. Deirtear "ag an doras" i gConnachta, Uladh agus sa Chaighdeán agus deirtear "ag an ndoras" sa Mhumhan. The latter is said in Munster.

The gender doesn't matter except with words beginning with 's', as explained above. I think your statement about t and d are also correct. (I'm assuming that I've got all the letters, rules and dialects right at the front of my brain -- If I've forgotten something then I hope someone points it out. Fáilte roimh cheartúchán ar bith.)