India is known as the Karma Bhoomi and it is generally said that our actions (including spiritual) bear the complete fruit if done only in India. I would like to know if our actions produce the same or different kind of results if we do not live in India and live in another country.

‘Karma’ in this context refers only to the Vedic Karmas (rituals prescribed in the Vedas). It is not applicable to Bhakti or anything else. You can be devoted to God and belong to any part of the world. When you do Nama Kirtan from anywhere in the world, it will give you exactly the same results as it would when done in India.

Why are the human avatars of the Lord centered around "Bharat" (India)? Did the Lord incarnate in other countries too?

In a house, we split the space into different rooms like kitchen, drawing room, Puja room, toilet etc. Considering the whole world as a house, India is the Puja room, and obviously the Lord lives there. Hence the Lord incarnates in this country alone.