In the UK, "pudding" is syonymous (if a little less classy) with "dessert". Other synonyms are "afters" and "sweet", though these are falling out of use. In the US, I believe, pudding specifically means a kind of blancmange dessert rather than dessert per se, so your sentences probably won't all work in AmE.
In BrE
(a) Yes, or "May I have some pudding, please" if you know there definitely is a dessert and you want some of it.
(b) Fine for everyday conversation - could sounds a little more tentative, as if you thought you might be refused.
(c) You don't need 'for dessert' - and the 'always' is a bit redundant if you're saying 'every day'.
(d) You have something for dessert after a/the meal. As above, in BrE you wouldn't say you have pudding for dessert - unless it was part of the name of a specific dessert such as rice pudding or summer pudding.