Do I necessarily have to use 第 when talking about a page number?
First, '书' in '打开书100页'(open book 100 pages) can be omitted

Yes, if you omitted '第' in '打开第100页'(open page 100) and wrote "打开100页", the sentence could be interpreted as " open one hundred pages". For this very reason, 第 cannot be omitted.