Well, the defeat of the Maratha confederacy was by no means the end of British military expansion. From the 1810s to the Mutiny of 1857, the British fought the Anglo-Burmese War, the Anglo-Nepalese War, two Anglo-Sikh wars, one Anglo-Afghan War, and the conquest of Sind by Napier. This does not take into account internal campaigns against the Pindaris and other smaller incidents.

The army of the East India Company was overwhelmingly dominated by soldiers of Indian origin. While European soldiers had their own distinct formations, Indian soldiers were the bulk of the infantry and cavalry, though the officers were all European. When the army was not fighting on the expanding frontiers of empire it was often seen as the principal safeguard against revolt. By 1856 there were as many as 200000 sepoys (indian soldiers) in the three armies of the East India Company - Bengal, Madras and Bombay.

As for transfer to the British Imperial Army, as far as I know, this would be rare , if not impossible for an Indian soldier. Moreover, most Hindus of the upper castes (comprising a majority of the Bengal army, which having 120000 soldiers was the largest) had strict ritualistic and customary restrictions against campaigning overseas. They were much more comfortable in the army of the EIC.

The details of the wars are available in any textbook of Indian history. For this issue, see Lawrence James, Raj: The Making and Unmaking of British India Hatchette, London (1997) For further details about the British Indian army, see the works of Kaushik Roy - War, Culture and Society in Early Modern South Asia, Routledge, London (2011) and David Omissi, The Sepoy and the Raj, Macmillan, London, (1994) - the first chapter.