Gaeilge: The Adjective Beag
Are these all correct?

capall beag -- a small horse

ainm capaill bhig -- a small horse's name

an capall beag -- the small horse

ainm an chapaill bhig -- the small horse's name

capaill bheaga -- small horses

ainmneacha capall beaga -- small horses' names

na capaill beag -- the small horses

ainmneacha na gcapall beag -- the small horses' names

Also, focloir.ie lists slender near plural nominitive adjective forms. Is the ending "aill" in "capaill" slender? Not sure if the extra l at the end cancels out the slenderness.

Additionally, how do we determine whether a noun is weak or strong so we know from focloir's chart which plural genitive form to use? Is a noun weak only if its nominitive and genitive plural forms (minus lenition or eclipse) are different? So in this case capaill (nominitive plural) versus capall (genitive plural) would mean that capall is a weak noun.

Question

Gaeilge: The Adjective Beag
Are these all correct?

capall beag -- a small horse

ainm capaill bhig -- a small horse's name

an capall beag -- the small horse

ainm an chapaill bhig -- the small horse's name

capaill bheaga -- small horses

ainmneacha capall beaga -- small horses' names

na capaill beag -- the small horses

ainmneacha na gcapall beag -- the small horses' names

Also, focloir.ie lists slender near plural nominitive adjective forms. Is the ending "aill" in "capaill" slender? Not sure if the extra l at the end cancels out the slenderness.

Additionally, how do we determine whether a noun is weak or strong so we know from focloir's chart which plural genitive form to use? Is a noun weak only if its nominitive and genitive plural forms (minus lenition or eclipse) are different? So in this case capaill (nominitive plural) versus capall (genitive plural) would mean that capall is a weak noun.

Answer

The first five are correct.
"small horses' names" should be "ainmneacha capall beag" (the adjective remains in the nom.sg. form when the gen.pl. of the noun is the same as the nom.sg., you got this right in the final sentence)
"the small horses" = "na capaill bheaga" (you got this right two sentences earlier).
"capaill" is slender, because the final consonant, "ll", is preceded by "i", this rule applies even to consonant clusters, the "i" makes the whole cluster slender: e.g. "iasc" - "sc" is broad; "éisc" - "sc" is slender ("na héisc bheaga"). Actually in Irish "ll" isn't considered a consonant cluster, it's a digraph (two letters representing a single phoneme or sound) and stands for a separate consonant distinct from that written with a single "l".
You can tell if a noun is strong or weak by looking at how it forms the plural:
If a noun adds any ending other than "-a" to form the plural, then it is strong (e.g. nouns forming the plural in "-anna", "-acha", "-í", "-ithe", "-ta", etc.); in strong nouns the plural is the same in all cases.
If a noun forms the plural by slenderising (the final consonant), or by adding "-a", then it is weak; in weak nouns the genitive plural has the same form as the nominative singular.
"Capall" forms its plural by slenderising, "capaill", so it is a weak noun, therefore the nominative plural will be "capall".