观察下列各式：

$(x-1)\div(x-1)=1;$

$(x^{2}-1)\div(x-1)=x+1;$

$(x^{3}-1)\div(x-1)=x^{2}+x+1;$

$(x^{4}-1)\div(x-1)=x^{3}+x^{2}+x+1;$

$…$

$(x^{8}-1)\div(x-1)=x^{7}+x^{6}+x^{5}+\cdots+x+1.$

$(1)$根据上面各式的规律填空：

$\;①(x^{2023}-1)\div(x-1)=$________________________$;$

$\;②(x^{n}-1)\div(x-1)(n$为正整数$)=$______________________.

$(2)$利用$(1)$中$\;②$的结论求$2^{2020}+2^{2019}+\cdots+2+1$的值.

$(3)$若$1+x+x^{2}+\cdots+x^{2021}=0$，求$x^{2022}$的值.