Yes, Indian English uses contractions as well. Their
use of certain tenses can sometimes differ though.
Contractions are standard in spoken English (they're certainly not just an American thing - I'm British and, as you can see, I use contractions as naturally as you do). Historically, contractions go back hundreds of years - even those who disapprove of their use, use them!
Contractions come from the rhythm and intonation of the language as well as the fact that English employs auxiliary verbs more to express what other Indo-European languages express in other ways (e.g. using modals instead of pronoun + adjective).
Pronouns (e.g. I, you, he, etc.) and auxiliary verbs (e.g. be, have, will, would, etc) are not stressed in English unless you're being emphatic, so often they are quieter/softer. In normal speech they clump together, often forming one word and one syllable, or they attach themselves to stressed words such as nouns.
In a normal unemphatic sentence, we emphasise nouns, adjectives, adverbs, and normal verbs so these words 'stand out' more.
Example:
"Ann is older than me" has the normal intonation:
ANN is OLDER than me.
The 'is' in the above sentence is often pronounced so slightly that it just sounds like an 's'. Hence in written English we represent this as ('s)
There are a few contractions that are American slang (e.g. 'gonna', 'gotta', 'oughta') which might not be used in India, but it's hard to judge with a world culture spread thru the global dominance of the English language. A lot of Americanisms are used quite widely in Britain among other English-speaking countries and even some non-English speaking countries (just look at the success of the American word OK).
Contractions can be problematic to people for whom English is a foreign language, but not if it's their native tongue so it depends on the individual and how they've come to know English.
(BTW. In formal written English, contractions should be avoided except perhaps "don't" and "can't" when these express an order not to do something.)