[phernikas]

Before answering your specific queries, let me give you the gist of the passage:
The Ummah was facing two enemies:
(1)A complex, well-armed one ie. the Mulhidin, who brought forth logical arguments to attack the Din
(2) the Hashawiyyah, whose simple arguments appealed to the masses and who inherited these beliefs from the Mujassimah of the Yahud and the Persians.

The Mu'tazilah are credited with defending the Din against these two groups. They first focused on the first and finished them off, then quickly sorted out the second.
However, their long debates with the first resulted in them being affected by some of the sicknesses of those they debated against.

Now for your queries:

And those who carried the burdens of those defenses were a group of the Mu’tazilah. So (now) they had become encircled by two enemies:
Who is "they"? The Ahlu Sunnah? The group of the Mu'tazilah who were defending the Din.

an occupying enemy from outside the nation with views and a philosophy upon which he was trained from an ancient time
Who is "he"? Is it the kuffar from outside the muslim lands? Yes

and a cruel enemy from inside the nation that the majority were on the verge of joining due to his asceticism while he was far from (grasping) matters of logic.
Are those the mutazila? These are the Hashawiyyah

The distortions of those who mislead from the Jews and the Dualists were broadcast over him.
Who is "him"? The Hashawiyyah

The extent of his work was the assault of the Theoreticians (Ahlun-Nazar) while not making a distinction between the enemy and the close friend. And if rule were given to him, he couldn’t defend (the nation) a single hour of a day.
Again, who is "him"? Whose "extent of work" is meant? Mutazila? The Hashiwiyyah ie. all they really did to defend Islam was to attack those who were defending it.

For this reason, these theoreticians were preoccupied by the former and they fell back from the latter until they completed the refutation of the crypto-infidels (zanaadiqah)
Who is meant with "the former" and who with "the latter"? Former- Mulhidin and Atheists
Latter- Hashawiyyah

exposed their distortions, and then demolished the argument of the Hashaweeyah and revealed the feeblemindedness of their views. And mental illnesses, a number of which cannot be held lightly, clung to the psyches of these theoreticians that were passed on to them from their debaters.
Who are those "debaters"? The Mulhidin that the Mu'tazilah debated with.