Hi Ng,
When we use 已经 in this context, we need the 了 there, because 已经 indicates some form of change happened. Thus, sentences 1 and 3 are not correct.
As for the others, while their usage is generally interchangeable, there is a slight difference in what they indicate. 2 and 5 indicate that he has already FALLEN asleep, whereas 4 and 6 simply indicate that he has gone to sleep but may not necessarily have fallen yet.
I hope this answer helps you. Please let me know if you still have any questions.