“Spent time for training” vs. “spent time on training” vs. “spent time on training of ”
Solution 1:
With "spend," I normally follow these patterns:
spend something + on + Noun
spend something + V+ing
So, I would correct all your sentences this way:
We've spent X hours on the training of the new employee.
We've spent X hours training the new employee.
I wouldn't use "for."
Solution 2:
No, and you might as well simply say:
We've spent x hours training the new employee.