Kathmandu: Solution

Pair up the words so that the two words in each pair can be preceded or followed by the same word to make a longer word or two-word phrase. Then find a third word which also makes a word or phrase with the same preceding or following word. This is highly ambiguous, but if you do it correctly then each third word is used twice:

         jumping \
         lumber  -JACK
         BLACK   <
         night   -GUARD
         trigger /

        /guitar
ELECTRIC-meter
        \CIRCUIT \
         ice     -BREAKER
         path    /

        /barrel
     GUN-powder
        \DOG     \
         rasp    -BERRY
         straw   /

        /pound
 QUARTER-staff
        >FINAL
    SEMI-quaver
        \skim

         cayenne \
         whole   -PEPPER
         GREEN   <
         kidney  -BEAN
         string  /

        /computer
   MICRO-scope
        >LIGHT
     SUN-beam
        \down

        /machine
    TIME-zone
        \WASTE   \
         flash   -PAPER
         manila  /

        /potato
   WHITE-wash
        \WATER   \
         not     -GATE
         star    /

Sort the third words alphabetically, concatenate them and extract the letters with the indices stated in the puzzle:

blaCKcirCuitdogfinAlgreenLightwastEwater

Alternate Solution from the Archives of Eric Albert, Author Unknown

RASP/STRAW = BERRY
LUMBER/JUMPING = JACK
==> BERRY/JACK = BLACK <==

COMPUTER/POTATO = CHIP
SKIM/WHOLE = MILK
==> CHIP/MILK = CHOCOLATE <==

POUND/METER = UNIT OF MEASURE
PATH/SCOPE = TELE
==> UNIT OF MEASURE/TELE = GRAM <==

STRING/KIDNEY = BEAN
MANILLA/CAYANNE = CAPITAL CITIES
==> BEAN/CAPITAL CITIES = LIMA <==

ZONE/MACHINE = TIME
TRIGGER/BARREL = GUN PARTS
==> TIME/GUN PARTS = MAGAZINE <==

GUITAR/ICE = PICK
NOT/NIGHT = WORDS THAT MAKE A HOMOPHONE WHEN YOU APPEND "K"
==> PICK/APPEND "K" = NIT <==

WASH/POWDER = BATHROOM NAMES
STAFF/QUAVER = MUSIC NOTATION
==> BATHROOM NAMES/MUSIC NOTATION = REST <==

STAR/FLASH = LIGHT
BEAM/DOWN = SUN
==> LIGHT/SUN = SPOT <==