Feminine nouns that start with s get a t-prefix after an whether there's a preposition present or not - a t-prefix is not an urú.
(except words that start with sc-, sf-, sm-, sp-, st- or sv-).
I'm not sure what you're asking.
There's only one genitive case in Irish, and the only genitive there is that masculine singular nouns are lenited after the definite article. Did you mean something else?
There are no vowels that cause lenition just because they're vowels.
The three 'a''s above are the Irish words for his, her, and their. a bhád (his boat - lenited), a bád (her boat - nothing happens, but if the noun had started with a vowel, it would add an h to the beginning: a hathair), a mbád (their boat - eclipsed)
I have a feeling that this isn't what you were asking, though. If you give some examples, I'll try again to answer your question.
A Cait48, a chara! a + séimhiú = his. a + (his) bád = a bhád = his boat, male possessor, good example, very helpful, GRMA!
a + h before a vowel = ( her? ) unhelpful as the example "bád" does not begin with a vowel thus it does not illustrate the point of when to add a 'haitch' (or 'an h') to indicate gender of possessor. Presumably, though not stated outright, "a" preceding a noun without séimhiú indicates feminine possessor?
a + urú on following word = their. a + (their) bád = a mbád = their boat. another good example, again GRMA! I've written all of this out in longhand. Now all I have to do is to remember it! Slan, Proinsias Ó Ciardhubháin.
"a + h before a vowel = ( her? ) unhelpful as the example "bád" does not begin with a vowel thus it does not illustrate the point of when to add a 'haitch' (or 'an h') to indicate gender of possessor." You add an h to any noun that starts with a vowel (a e i o u) when you use 'a' meaning 'her.' I left it up to the reader to pick one. If none came to mind, here's one: a haintín (her aunt). Also, there's no need to ask "her?" because 'her' is already written at the end of the line for you.
"Presumably, though not stated outright, "a" preceding a noun without séimhiú indicates feminine possessor?" Nope. Not stated because that's not what happens. There are plenty of words that start with a consonant that cannot be lenited ( = no séimhiú) or eclipsed ( = no urú) -- 'nathair' (snake) for example. 'A nathair' could mean 'his snake,' 'her snake,' or 'their snake.' In such cases, you need to rely on context or use 's'aige' (his), 's'aici' (her), or 's'acu' (their) after the noun. Ex: an nathair s'aici = her snake.
You add an h to any feminine noun that starts with a vowel (a e i o u).
You prefix an h to any noun, feminine or masculine, that starts with a vowel (a e i o u), after the 3rd person singular feminine possessive adjective adjective a ("her").
an t-úll - masculine
a úll - "his apple"
a húll - "her apple"
an ubh - feminine
a ubh - "his egg"
a hubh - "her egg"