Meta: Since the trees & guards function separately from the pit (and reaction to the pit), I'd key them as two separate entries. But carrying on with just the one assuming we have to stick with the caves of chaos map as-is, and trying to make it shorter...
A. Kobold Lair: This cavern entrance is nestled at the edge of a large stand of trees. Weeping branches hang low over the lintel; medium creatures will have to brush past them or duck under them.
Specify the height of the branch tips, and you can eliminate the whole portion after the semicolon.
The tunnel beyond is unlit.
Probably don't state this. If you must, move "unlit" into the first sentence.
2/6 chance that 8 kobolds lurk among the trees. (AC 7, HD 1/2, hp 3 each, #AT I, D 1-4, MV (40’), Save NM, ML 6)
Interesting interpretation that 2:6 represents the kobolds' existence, as opposed to the boldness of always-present kobolds. Could go either way, I guess.
10’ pit inside entrance (marked on map). 3/6 first rank, 1/6 second rank, 1d6 damage, hinged lid closes to prevent escape.
Don't need to state the damage. (Insert counter-argument here about this being a beginner DM's module.)
Detect automatically with probing.
"Probing detects," or add wording earlier that otherwise implies the simplicity of design and specific trigger condition. Maybe "camouflaged lid" and triggers with 20 lbs. But that starts to get long again.
Kobold guards may lurk in the underbrush around the cave entrance; if present they carry 1d8 sp each.
Move the gold to their stat block (which uses abbreviated form, so shorter already), and you don't need to say, "if present."
The entrance itself is braced by a heavy stone lintel on stone pillars, set into the earth. Beyond lies a 10’ high tunnel roughly chiselled from the living rock.
Combine the sentences: "A heavy stone lintel and stone pillars brace the 10 ft high entrance into chiseled living rock."
There is an intersection 30’ into the tunnel; the din of conversation from area 6 can clearly be heard, and prevents any attempt to hear noises from areas 1 or 2.
I'd key this separately, but again: Can't change the map. Recombine: "At 30 ft in: Noise from area 6 emanates, preventing hear noise from areas 1-2." (Maybe that's shorter than you want in this section though.)
The intersection is protected by a pit with a hinged metal lid, camouflaged to look like stone. Tapping the metal lit will reveal its presence. After the pit is triggered, the lid will close and latch from the outside. The pit mechanism is noisy and will alert the creatures in areas 1 and 2. The kobolds in area 1 will be alerted by any light source, but will take no action until they know if the party will fall victim to the pit trap.
Combine details, switch to active voice, remove "will" constructs, and drop the "light source" part (since it's not in the original):
"A camouflaged pit with hinged metal lid protects the intersection. Tapping reveals it. When triggered, the lid closes and latches from outside; the noise attracts the creatures from areas 1-2."
Could even remove the "and latches..." part because it's not in the original.
This might all defeat the purpose of what you were trying to do though...