Do I necessarily have to use 第 when talking about a page number? Can one say 打开书100页 or is it necessary to add 第 before 100?

I was thinking that it would be the difference between saying "open the book at/to page 100" (with 第) and "open the book 100 pages" (without 第, which sounds wrong to me).
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.