I know it sounds a stupid question but in English comics, if someone is calling a woman named "Laura" from afar, you write "Lauraaaaaaa", with the last letter repeated. How do I do the same with "Катя" in Russian? If I write "Катяяяяяяя" it seems like I am saying "Katyayayayaya" instead of "Katyaaaaaaa". How do you do that?
If you write in Russian, you can write "Катяяяяя". No one would read it as "Катя-йа-йа-йа". "Катяаааа" looks unusual but can be used in situation when the last sound is transformed in a shout like "Катяяяааааааа".
If you write in English it's better to write "Katyaaaa".
Note that in Russian often the stressed vowel is shouted long too, not only the last one - so you can also write "Каааатяяяя" or "Владииииимиииир".
That depends on whether you want the literary norm or something informal you could use, say, on the internet.
The literary norm would be Ка-а-атя-а-а. Note the hyphen before each vowel repetition. I'm surprised no-one else brought it up. To answer your specific question, the modifying letters do revert to their non-modifier counterparts in repetitions (hence Я-а-ана and Ю-у-ур!). Keep the number of repetitions down to one or two. Three or four would mean it's really drawn-out. More than that means it's a children's story or the writer's got an amateurish fondness for typographic mannerisms.
In informal writing, Кааатяяяя is perfectly fine and you can hold down the key much more liberally.
Well, actually I'm surprised with the answers given. As a native speaker I'd rather will stretch the stressed vowel, something like "Ау, Каааааааатя! Ты где?"
The same rule will apply to any name.
So now let's consider what happen if
я is stressed, then actually indeed something like "Ну Яяяяяяян, ну чего ты копаешься". So я (or ю - Юююююр) does not usually turned into "йааааа" (ой "йуyyy") - it's just that it's pronounced that way.
'Ё' is slightly different story, it's quit often ommited but still you'd rather see "ёёёёёёжик" or just "ееееежик" but never "йоооожик".