Great question! Of course this is a huge time-span, covering the Delhi- and Bahmani-Sultanates, and possibly some of the smaller Islamic states that followed. As far as I know, similar to Muslim rulers in other regions, in the Sultanates there were religious liberties (usually connected to special taxes payed by non-Muslims), and local elites from various religious groups were integrated into the administration, providing much-needed support to the foreign rulers. Furthermore, Mughal rulers like Akbar (probably outside the time-frame you're asking about) are known for the atmosphere of cultural freedom at their courts, with Akbar ordering a translation of the Koran in Persian and sponsoring magnificient books produced by Persian, Muslim and Hindu artists. There was not only tolerant action, and mass-conversions and mistreatment of other religious groups were noted, but these are usually described as exceptions. Because of the large time-span and territory I'll try and narrow it down to an example I know a bit about more, concerning the second part of your question about independent kingdoms.

The Vijayanagara-empire developed into the largest historical realm of South India and existed in varying forms from the mid-14th to the mid-17th century. It came to existence following the drawing back of the Delhi-sultanate from South India and the demise of earlier states in the Deccan. The empire was ruled successively by four dynasties: The Sangama (~1350-1486), Saluva (~1486-1505), Tuluva (1505-1569) und Aravidu (~1569-1654)-dynasties. Its huge expansion was made possible through a strong military, focusing on a cavallery that was connected to the employment of Muslim mercenaries. With the third dynasty, the Tuluvas, and especially under Krishnadeva Raya the state reached its political and cultural high-point, with administrative reforms, temple buildings, and patronage of poets and scholars alike. Krishnadeva triumphed over the troops of the sultanates of Central India, whose sultans he then reinstated. However, under following rulers Vijayanagara lost power, and after the battle of Talikota in 1565 the capital was looted. Over the following 100 years the Aravidus ruled successively from three different capitals over ever-diminishing territories.

Looking more specifically at religious aspects, one of the rulers' titles, “Sultan among Hindu-kings” for Bukka I., already shows that Vijayanagara was part of a South-Indian political landscape strongly tied to Islamic political traditions. The systematic religious politics of the kings can also be seen as an influence on their success. Thereby, individuals could take part in different spiritual communities at the same time, and there was no royal control of religious authorities. As the population prayed to a diverse array of deities, e.g. local gods, natural sites, as well as shrines of Muslims and Jains, various very different shrines and temples were built in the realm. Furthermore, elements of South-Indian architecture were mixed with Indo-Islamic traditions. This style was confined to public and royal buildings in the capitals, in connection to their representative function regarding Muslim envoys. Images of all main deities from different parts of the empire were also integrated for worship into temples of the capital.

In accordance with the 'realpolitik' situation, whereby the realm was surrounded by various powerful Islamic states, one can notice Vijayanagara's adaption to Islamic culture, changing its Hinduistic customs in the process. Another influence was manifest in the trade with Muslim states via the Indian Ocean. A more concrete and fascinating form of this inter-religous contact, next to architecture, rulers' titles, military techniques and administration was the clothing at court – royal clothes and textiles represented royal authority. Both central pieces of clothing at court were derived from Islamic styles: A cone-shaped cap, kullayi, came from the Persian kulah that goes back at least to the 15th century. A long robe, the kabayi, was connected to the Arabic qaba, common in Arabic regions already in the 7th century. Both items stood in radical contrast to traditional Indian court styles, where men usually wore no upper garments or simply put on a scarf. We can see here how nobles of the court adapted to customs in accordance with Islamic norms by wearing upper-body clothing. In addition, royal clothes were ritually passed on to foreign diginitaries and lower nobles. This ritual goes back to ceremonies at the courts of North Indian Islamic states and was not in use in South India before Vijayanagara. These gifts were political acts, showcasing the legitimacy of the giver and the dependance and prestige of the receiver. In this way, court rituals were made understandable in the political lingua franca of the Islamic world, as to be accepted by Islamic rulers.

To sum up, one can see in the different forms of adaptation and acculturation a deliberate act from Vijayanagara's court elite and rulers, directly connected to changes in the political culture at court, and in the political situation of South India. Hope this helps! I'll add some literature soon when I have better access.

Sources: For a good overview on Vijayanagara see Rao, Nalini, Royal Imagery and Networks of Power at Vijayanagara – a Study of Kingship in South India, Delhi 2010.
More focused on the political situation and religious identity there: Sinopoli, C. M.: ''From the Lion Throne'', JESHO Vol. 43, 2000, p. 364-398. (if you have jstor access)
A detailed look at court ceremonies, clothing and art is Wagoner, Philip B.: ''´Sultan Among Hindu Kings´: Dress, Titles and the Islamicization of Hindu culture at Vijayanagara'', in: The Journal of Asian Studies 55/4, 1996, p. 851-880. (again on jstor)

Edit: Added literature.