The shifter has to be in a certain position (the position where the cable would be at the lowest tension) to be able to push the cable end (head) out of the brifter. Otherwise, the ratcheting mechanism is rotated such that the cable end is trapped inside of the brifter. Also, You'll probably need to have the brake cable disconnected to be able to have the top of the brifter rotated far enough to easily get at the cable end.
The cable slides inside of the housing so yes, you can push the cable out of the housing. Not sure if that answers your question or not.
The silver cylinders sound like tension adjusters. If you don't have tension adjusters mounted on your frame, then those must be inline adjusters. Small adjustments typically won't make much of a different if your bike is shifting properly anyway. Turn the adjuster a full turn and you'll likely cause some problems, or fix them if you know what you are doing
