Both can be used (with a few corrections). Though to me, and this is only my personal opinion, there are slight differences in meaning.
'My employer sent me abroad several times.' Suggests that the action is complete, you are unlikely to be going abroad again.
'My employer has sent me abroad several times.' Suggests that the action may reoccur, you may go abroad again.
For the second example, I would simply say, 'I was "Employee of the Month" twice.' Or, 'I have been "Employee of the Month" twice.' I wouldn't use 'became/become'.
Like I said, these are just my personal opinions, perhaps a teacher will have a different or more useful interpretation.
Student or Learner