He's an extremely important character, but you get more of him and his backstory in Season 2 (which is also when the other family member, Azula, shows up). In Season 1, he seems to mainly be a standoffish mentor who really just wants to enjoy retirement, but by the end he's a major player and the focus of what was arguably the most emotional scene of the series entire run. (They did have to take him off-stage for a little while when his voice actor died.)
From what I've heard of this film, they did touch on all the major notes (specific events) but put the wrong spin on just about all of them (getting rid of all the humor, which wasn't silly in the show, it showed that although these kids were forced to deal with adult responsibilities they were still kids, and so the humor became poignant; giving all of the girl's triumphant moments to boys -- battles, heroic speeches -- and writing her as losing fights that she'd won in the series; letting all the major happenings be covered by voiceover; toning down the impact of the violence.)
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From what I've heard of this film, they did touch on all the major notes (specific events) but put the wrong spin on just about all of them (getting rid of all the humor, which wasn't silly in the show, it showed that although these kids were forced to deal with adult responsibilities they were still kids, and so the humor became poignant; giving all of the girl's triumphant moments to boys -- battles, heroic speeches -- and writing her as losing fights that she'd won in the series; letting all the major happenings be covered by voiceover; toning down the impact of the violence.)