Hello, the short answer to your question is yes, there certainly were attempts by other colonizing powers (most notably the French) to exert colonial control in India. I wrote a short paper for a course called "Modern British Empire" in undergrad that covers the broad strokes of several of your subsidiary questions that I will paste below, I hope it can be of some help!

Of all of the British Empire’s imperial projects, the conquest and subjugation of India by the East India Company stands out as perhaps the most ambitious expansion of British power throughout the empire’s history. The East India Company’s evolution from a small commercial operation at its founding in 1600 that sought to deal directly with spice suppliers in the East (James, 24) to its role as an established, apolitical, and highly profitable commercial enterprise along the Indian coast in the 1740s (James, 123), set the stage for the Company’s concerted efforts at expanding profits and seeking out wealth through whatever means necessary. It was this insatiable drive for wealth by predatory, profit seeking imperialist officials and generals (James, 124), coupled with competition and the perceived threat of the French colonial presence in India (James, 124), that motivated British expansion across the sub-continent.

The “Treaty between the Nawabis and the Company of 1765” is a prime example of the motivations that governed British imperial expansion into India throughout the rest of the 18th century and the early 19th century, and it provides an example of the depth to which the East India Company delved into the complex local politics of the crumbling Mughal Empire and its rulers after the 1740s. The treaty was an agreement between the Nawab Shujau-ul-Dowla, of Oudh, the Nawab Najmu-ul-Dowla of Bengal, and the East India Company that promised the establishment of “a perpetual and universal peace, sincere friendship, and firm union” (Treaty, 28) between the contracting parties. The treaty was a defensive pact and alliance between the Company and the Nawabs, or princes of Oudh and Bengal, with subsequent articles outlining the handsome revenues and payments that the company was to reap in the uneven terms of the treaty, which traded Company protection and support to the princes in exchange for the surrendering of revenues and authority (James, 129).

The Company’s extensive involvement with Nawabs came about as a response to the broader global, imperial rivalry between the Britain and France, and the wars that their rivalry produced. As the War of Austrian Succession raged in Europe from 1740 to 1748 and expanded globally throughout colonial holdings (James, 124), both the French Compagnie des Indes and the East India Company began to actively meddle in Indian dynastic politics in order to gain the upper hand over the other (James, 124). Both the French and British Companies sponsored rival princes with troops by 1750 (James, 126), and the colonial arms race between the French and the British turned into a proxy war between the two imperial powers, who sent professional troops and navies to ensure they retained their foothold in India (James, 127). As war broke out in Bengal in 1756, the connection between war and profits became more and more apparent, and “civilians and soldiers discovered that the profits of war far exceeded those of trade” (James, 129).

This explicit connection between war and Company profit is central to the terms of the “Treaty between the Nawabs and the Company,” which was part of a broader process of “expansion by coercion and treaty” (James, 129) that came to characterize the increasingly uneven power relations between the British and their Nawab counterparts. In Article Two of the treaty, it states that “In the case of the English Company’s forces being employed in His Highnesses’ service, the extraordinary expense of the same is to be defrayed by him” (Treaty, 28). With all of the financial burden being placed upon the Nawab of Oudh in case of any conflict, the Company and its troops profited immensely from such an arrangement, and ironically, the Nawabs allied with the East India Company were essentially financing their own subjugation.

Members of the Company of all ranks, from the leadership to the common foot soldier, profited handsomely from such arrangements, in which grateful princes bestowed vast wealth upon top Company officials, in addition to paying the salaries and expenses of the British troops who established garrisons within their cities for “protection” (James, 129). This very process is described in Article Ten of the treaty, which states that “English forces shall be withdrawn from the dominions of His Highness, except such as may be necessary for the garrison at Chumar, or for the defense and protection of the King…” (Treaty, 30).

The increasingly administrative role of the Company in governance within India, and the revenue it derived from traditional tax collection that princes conceded in the treaty and others similar to it is found within Article Five, in which the Nawab of Oudh is given leave to rule certain districts “on condition of his paying the same revenue as heretofore” (Treaty, 29). The vast wealth of the princes and their politically fragmented empire, and military weakness in the face of superior European weaponry presented the Company and its members with the prospect of continued personal enrichment, and served the empire simultaneously by expanding British control further inland, while denying France and other imperial rivals the opportunity to do the same.

Articles Six and Eight of the treaty provide further evidence of the powerful monetary incentives for expansion and greater Company control over India and its rulers. Article Six lays out highly specific terms for the remuneration of the Company “in consideration of the great expense incurred by the English Company in carrying on the late war” (Treaty, 29-30), and sets up payment to the Company in installments over the course of thirteen months. These payments, and their highly structured and detailed terms, including payment via rupees and jewels in set installments demonstrate how war and coercion became systematic tools for Company gains. Military power became political power, and political power quickly translated into financial gain. Article Eight, which “allow the English Company to carry on trade, duty free, throughout the whole of his dominions” (Treaty, 30) is another example of the extent to which the treaty of “sincere friendship” was truly a one sided affair, in which the East India Company pressed its military advantage and power by ensuring that the Company itself remained above taxation throughout the Nawabs’ territories.

The transference of the power of taxation, and the Company’s exemption from duties itself also reflect shifting power dynamics, in which on the surface, the Nawabs retained their sovereignty and their lands, but remained as figureheads. Without the power to tax, and with vast portions of their wealth flowing into East India Company coffers and pockets, the princes entered a vicious cycle in which the process of coercion and treaty continued, and tightened the grip of the Company upon key administrative institutions, such as taxation, that drained both money and power away from the heirs to the Mughal empire, and strengthened the British grip upon what would come to be the jewel of the empire.

Source: Lawrence James, “The Rise and Fall of the British Empire,” (New York: St. Martin’s Press, 1997)