I am not a teacher, but I do not understand why you can't say 'I am Indian'.
I learnt that the infinite article was not used with the names of countries aside from the USA (composite name of country).
About your other quote, I think the right sentence is 'I am taking a bath' without article is maybe used in speaking English but for me it seems to me informal to avoid the use of the indefinite article. It is like 'I have a beer'.
Do not forget I am just a learner. (Sorry for my mistakes).