I am not sure what you mean by theoretically and practically. If you mean the loyal warrior protecting defenseless farmers and following his lord to death, as pointed out here and here and here, those are more the exception than the rule. And since society was in chaos, I'm not sure how much we can talk about "one's role" either.

So we are in the Sengoku. The term 大名 (Daimyo) is not a formal title yet. The term comes from the later Heian period that meant "Lord of a big manor". As opposed to a small one. Well that's not very specific. In the Edo the term will be formalized to mean lord of a domain with at least 10,000 koku. But that's in the future.

As "Daimyo" is not a helpful term, we should look at three terms. First are the Protector (守護 shugo) and the Deputy-Protector (守護代 shugodai). On paper these are the people in charge of one of domains of Japan. The Protector ruled, and his Deputy did when the Protector was away, often in Kyoto. That was the system of the Muromachi Shogunate (which we technically are still in). But this is the Warring States. Those titles are only as good as the legitimacy it conveys. The third type of person are the Men of the Province (国人 kokujin). These are samurai with their own manor holdings who may or may not listen to the shugo and/or his shugodai. Likewise the shugodai may or may not listen to the shugo. A “daimyo” is usually a shugo (back in the peaceful days a shugo was a daimyo), occasionally a shugodai, who has de facto control over at least a large part of one province and answers to no other samurai lords. Many (but a minority) daimyo’s domains spanned multiple provinces.

So one thing you can be sure to happen is that there will be war. The situation is a might-makes-right free-for-all. Some shugo clans successfully unites their provinces. Others became puppets for their shugodai to be eventually replaced. Even kokujin could rise up to become shugodai and shugo. This process of lower status samurai replacing their lords is called Gekokujo (下克上) which means exactly that. There will also be inter-province wars. By 1551 a handful of daimyos (whichever title they actually started out as) had consolidated their province and lead expeditions to neighbouring provinces. But that doesn’t mean their hold on their own provinces are solid. Rebellions still happen. In 1551 the Ōuchi Yoshitaka, a fairly successful multi-province daimyo up to this point, met his end when one of his shugodai betrayed him. And a lord always need to watch out for them and/or his uncle/brother/nephew/son plotting to overthrow him. In 1548 Oda Nobuhide, shugodai of Owari Province, had to quell a rebellion by his nephew. His son Oda Nobunaga succeeded him in 1551 and began fighting with his younger brother over succession. And not all daimyo had unified their province. In fact, many (most?) haven’t. Nagao Kagetora (better known by his later name Uesugi Kenshin) unites Echigo that year. The Date up north, Chosokabe in Shikoku, and Shimazu in Kyushu are still trying to unite their provinces.

Lord of the Sengoku could be expected to be warring almost every single year. So what did daimyos do when they’re not fighting? Well preparing to fight is important. Perhaps ordering a few hundred arquebuses from some artisan group. Arranging marriages with the shugo/shugodai/kokujin are also important to improve relationships, secure alliances or at least peace. In 1550 Takeda Harunobu (Shingen)’s older sister, Imagawa Yoshimoto wife died, resulting in a blow to the alliance between the two clans. In 1554 the two of them decided to cement their alliance while bringing Hōjō Ujiyasu into a three way alliance-marriage, which Takeda Shingen broke as soon as it suited him in 1567-68. Also we can’t forget other political things, like discussing strategy and rewarding your vassals, petitioning the Kyoto court to formally recognize you as shugo (Nagao Kagetora in 1550? 1551?), etc. Perhaps construct or expand on this or that castle at strategic points. Of course ensuring prosperity and peace in your province is also important, if for no other reason but less unrest, more tax, and more men to levy. Things like land surveys, improving the roads (Mori), and building dykes to control flooding and improve irrigation (Takeda), establish and improve mines, and promoting the nanban trade (or just trade in general) are all things daimyos did (or ordered others to do, I suppose). Besides these things, due to the social breakdown all over Japan many daimyo passed some basic law codes over their domains. Establishing temples seem to have been common too.

As for art and pass times (not my specialty), falconry, hunting, sumo wrestling, nō theatre, tea ceremony, and renga poetry were popular. Collecting paintings, painted folding screens, tea bowls, and weapons seem to be popular too, as hobby, status symbol, and objects that could be used to reward vassals.