有没有begin doing sth？

有，和begin to do sth意思完全相同
start也是这样用，加doing和to do都对

有..
begin sth. / begin with sth..
begin to do./
begin doing sth.

eg: She began working last year. 这句子没错.

有.begin to do sth. 和begin doing sth.都有.朗文给的例句是
We began to wonder if the train would ever arrive.
我们开始怀疑火车到底会不会到.
I left teaching in 1990 and began working in my present job.
我1990年离开教学岗位,开始现在的工作.
可以begin doing sth.中的动名词多为延续性动词,表示一种状态或行为的开始.

有
Begin asking questions; begin to speak
开始问问题；开始说话

有.

有.