1/ "9月底的預產期。因此邀請國內母親首次來英探親照顧。"
I hope you didn't disclose this information on your invitation letter, and didn't use it as your reason to invite your mum to England. Did you do that?
2/ "6月初在廣州簽證處遞交的申請,遭拒(大概因為我申請的是一年多次往返簽證)。"
In good practice, when your mum's visa was denified, the officer would give exact reason. What was it? "大概因為我申請的是一年多次往返簽證" is only your guess. It shouldn't be used as an excuse not to award your mum's visa. If your mum is not sure of the reason, she can ring them to find it out.
3/ "再次申請獲簽的幾率大嗎?"
It depends. That is why I have asked you to figure out the reason of denying the visa. Next application should avoid the same issue, it doesn't necessarily mean to lie.
4/ "隔多長時間遞交申請合適?"
No definite answer to this one. Your mum can reapply the next day.
Good luck.