I have absolutely no idea what "beer coils" are in relation to draft lines. If you're talking about a cold-plate, like you'd use for a jockey-box for event serving, you're off-base for draught beer delivery systems.
The draught lines must be chilled for their entire length. Failure to do so will result in the beer "breaking" in the lines--going above the CO2 saturation temperature. Cooling this at the tap will not solve the problem--you'll be serving foam.
For long draught lines, use an insulated, glycol-cooled beer trunk. Micromatic and Banner are both good sources for this, and the dedicated glycol chiller that goes with the beer trunk.
You're right about the 3/8" line for the trunk. A length of 3/16" "choker" line at the tap compensates for the delivery pressure to give you good pours. The length of the choker will depend on your beer carb level, how long the lines are, lift from the keg to the faucet, and elevation. Find the Draught Beer Quality Manual online and download it.
Terminal Gravity Brewing,