There are currently 2136 常用漢字. However, using writers will often use 漢字 outside of this list.
So I ran a script to calculate the total number of unique 漢字 in Harry Potter and the Chamber of Secrets (a modern novel mentioned there) and on ノルウェイの森/Norwegian Wood (a "famous" Japanese language novel, though perhaps not "typical") and compared it to the 常用漢字 list. You can see some of the code (it just has the results for the distinct values in the books) here. Here's what I got
Harry Potter and the Chamber of Secrets: 206 non-常用漢字 out of 1768 漢字 (肱 might be a typo for 肘, 躯 for 体, 荊 for 茨, and 綴 for 絞. I also wonder if 姐 and 嬉 are old forms or my source text has typos. I also have no clue about 杵 and 磔, which are likely wrong. I'll try to confirm with a physical copy in the future.)
{'蘇', '捌', '曾', '鈎', '撫', '剥', '捷', '篭', '恰', '鍮', '撒', '涎', '虱', '樽',
'炸', '塵', '椀', '吼', '淀', '絢', '歪', '掠', '瞥', '揉', '眩', '董', '錆', '嘘',
'楊', '蝶', '靄', '疼', '呆', '叱', '弛', '攣', '棍', '猾', '藁', '軋', '荊', '揃',
'纏', '甥', '這', '樫', '苺', '掻', '惚', '杖', '煤', '咳', '煽', '辻', '瓢', '詫',
'捺', '脛', '溢', '溜', '蝋', '鎧', '糞', '芻', '腿', '噛', '唸', '痺', '鍔', '霞',
'狡', '翠', '昂', '罠', '燭', '蜘', '牢', '躯', '叩', '杵', '雛', '佇', '鳩', '呟',
'灼', '爛', '逢', '嘴', '黴', '呑', '兎', '嘩', '濡', '轟', '鞄', '咎', '棘', '笥',
'壷', '閃', '猜', '楓', '聡', '麿', '箪', '捧', '馴', '藹', '蛙', '肱', '馳', '覗',
'釘', '茜', '埃', '愕', '雀', '皺', '躇', '鋸', '踝', '穣', '奢', '檻', '箒', '姐',
'甜', '嵌', '萄', '螺', '嬉', '唖', '秤', '仔', '掴', '窪', '狼', '翔', '綴', '腔',
'箔', '芍', '躊', '鮭', '磔', '鱗', '悶', '禿', '躓', '蛛', '惹', '閂', '哮', '瓜',
'腑', '曙', '槌', '沫', '狐', '倦', '卿', '剃', '栗', '燻', '頷', '憑', '斧', '蟻',
'橙', '竿', '怯', '紐', '穢', '耽', '尖', '咆', '凱', '蒼', '肋', '錘', '踵', '窩',
'縞', '竪', '痙', '鞭', '吊', '喧', '聳', '髭', '焚', '廻', '斐', '堵', '頬', '醇',
'瘤', '鋏', '葡', '〇', '鉤', '鷲', '漉', '舐', '滲', '堰'}
ノルウェイの森/Norwegian Wood: 167 non-常用漢字 out of 1746 漢字 (Like before 姐 and 嬉 might be old forms/typos. I'll try to confirm with a physical copy in the future)
{'梯', '遙', '鱸', '鎧', '梢', '徨', '撞', '喧', '昂', '雀', '而', '鳩', '庇', '轟',
'燈', '葡', '拗', '函', '禿', '悶', '槌', '訊', '紐', '碗', '鞄', '鮭', '蝕', '蛙',
'軋', '舐', '檻', '湘', '茗', '萄', '蓮', '鯵', '綺', '裳', '癬', '胡', '吠', '縞',
'詫', '洒', '樵', '奢', '掻', '痰', '爛', '呑', '姐', '錆', '疥', '殆', '糊', '〇',
'駕', '罠', '夾', '頬', '淋', '蝿', '楊', '鴉', '歪', '篇', '睨', '吊', '疼', '叩',
'嚢', '噛', '咳', '睾', '竪', '唖', '菱', '濡', '囁', '仄', '糞', '蹙', '脆', '稜',
'彷', '褪', '漕', '蟻', '曝', '栗', '釘', '杖', '柏', '螢', '甥', '旭', '剃', '耽',
'惣', '愕', '癌', '喰', '掴', '醤', '嘘', '嘔', '顰', '腿', '疇', '瑞', '蹟', '嬉',
'馴', '蝉', '垢', '鯛', '桐', '弘', '挪', '嘩', '釆', '揃', '云', '撫', '玲', '屑',
'噌', '怯', '噂', '這', '桶', '鰻', '髭', '迂', '蛾', '蝶', '瓜', '巳', '鱈', '苔',
'苑', '綜', '瞞', '墟', '阿', '辿', '茫', '睥', '茸', '樫', '蠅', '鼠', '弛', '坦',
'穢', '倦', '鹸', '椀', '凌', '眩', '伊', '鴨', '叱', '煉', '苺', '捧', '劫'}
If we include 人名用漢字, which brings the total 漢字 to 2999, Harry Potter and the Chamber of Secrets has 104 that aren't in either list and ノルウェイの森/Norwegian Wood has 78.
{'姐', '呟', '壷', '嘘', '鍮', '咆', '呆', '爛', '滲', '瘤', '掴', '肱', '咳', '憑',
'穢', '髭', '叩', '禿', '塵', '吼', '腿', '炸', '痺', '篭', '皺', '猾', '噛', '蜘',
'藹', '舐', '腑', '頬', '甜', '揉', '黴', '芍', '愕', '煽', '罠', '閂', '聳', '檻',
'眩', '燻', '鈎', '嵌', '吊', '痙', '唸', '糞', '咎', '捌', '歪', '攣', '剃', '脛',
'蟻', '踵', '荊', '虱', '怯', '蛛', '奢', '蝋', '棍', '靄', '掻', '窩', '〇', '佇',
'頷', '躯', '哮', '笥', '躊', '漉', '躓', '箒', '叱', '嘴', '芻', '鍔', '剥', '鋏',
'躇', '埃', '箪', '涎', '棘', '疼', '呑', '狐', '猜', '鉤', '踝', '狡', '鮭', '磔',
'唖', '軋', '悶', '覗', '蛙', '牢'}
{'拗', '囁', '褪', '彷', '唖', '蹙', '眩', '鰻', '睥', '蠅', '嘔', '愕', '徨', '禿',
'鮭', '咳', '疥', '鱈', '髭', '疇', '蛙', '釆', '癌', '醤', '洒', '顰', '鼠', '樵',
'〇', '蝕', '鹸', '掴', '呑', '爛', '癬', '螢', '吠', '吊', '蝿', '頬', '悶', '睾',
'蟻', '糞', '叩', '怯', '掻', '疼', '茫', '夾', '穢', '垢', '叱', '蛾', '鱸', '歪',
'鯵', '睨', '軋', '瞞', '舐', '蝉', '罠', '茗', '脆', '奢', '腿', '挪', '嚢', '檻',
'鴉', '仄', '嘘', '噛', '痰', '姐', '墟', '剃'}
As for your second question as to how many 漢字 you need to know to know 99.7% of the ones used, I used the results of this Github repository to get the following:
Aozora Bunko (there are many many "old" stories): 3820
News (June 2014 - June 2015, more samples from 2015, from 4 different sources. Only article titles, subtitles, main text body and image captions were scanned. Weather forecasts and area-specific news were not included). : 2007
Wikipedia (a wikipedia dump: 2835
Twitter (steaming data for 1 week): 2312
Also according to this analysis of 朝日新聞朝夕刊's paper through the entirety of 1993, you need to know 1977 to be able to understand 99.7%.