suggest sb doing sth這種用法對嗎？

用法是對的

suggest doing sth建議做某事（動名詞作賓語）

suggest sb's doing sth（動名詞作賓語帶有自己的邏輯主語sb's：所有格+動名詞，叫動名詞的復合結構)

suggest sb doing sth（動名詞的復合結構作賓語時，可以用代詞賓格或名詞普通格代替）

例子：

I suggested his leaving soon.

=I suggested him leaving soon.

I suggested the boy's leaving soon.

=I suggested the boy leaving soon.

suggest sb.do sth和suggest sb.doing 有什么區別？

suggest sb.doing 建議做某事：是suggest 后加sb作賓語，doing 是賓語補足語,是動名詞的復合結構作賓語。

suggest sb.do sth 建議某人做某事：是suggest 后加了個賓語從句。that 不做成分，所以省略了that 這個句型。