Snowman Bowling

Snowman 1.0.2, 157 bytes

(()("789:045600230001"4aG::48nSdU][:#:]eq]/nM;AsI[:"_____"wR["     "wR/aC;:"( : )"wR["(.,.)"wR/aC;bI;:"  "wRdUaC;bI\#**\;aMaZ:" "aJ1AfL;aM;aM1AfL"
"aJ1AfL*))

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When I saw this challenge, I knew I just had to answer in the perfect language...

This is a subroutine that takes input as an array of numbers and outputs as a string via the current permavar.

Wrapped for "readability" / aesthetics:

(()("789:045600230001"4aG::48nSdU][:#:]eq]/nM;AsI[
:"_____"wR["     "wR/aC;:"( : )"wR["(.,.)"wR/aC;bI
;:"  "wRdUaC;bI\#**\;aMaZ:" "aJ1AfL;aM;aM1AfL"
"aJ1AfL*))

Slightly ungolfed / commented version:

}
1wR`
3wR`aC`
5wR`aC`
6wR`aC`
9wR`aC`
*

((
    )(
    "789:045600230001"  // pin layout data
    4aG                 // split into groups of 4; we need each row twice
    :                   // map over groups of 2 output lines
        :               // map over pins (or whitespace)
            48nS        // subtract ascii '0'
            dU][        // duplicate the pin; we need it in the if{}
            :           // if (pin) {
                #:]eq]/nM;AsI[:"_____"wR["     "wR/aC;:"( : )"wR["(.,.)"wR/aC;bI
            ;:          // } else {
                "  "wRdUaC
            ;bI         // }
            \#**\       // maneuver the permavars around to discard pin
        ;aM
        aZ:" "aJ1AfL;aM
    ;aM
    1AfL                // flatten (simulate a flatmap)
    "
"aJ                     // join on newline
    1AfL                // flatten again into a single string
    *
))

#sP

stacked, noncompeting, 118 bytes

I added deepmap and a few other things after this challenge, along with tons of bugfixes. Try it here!

@a((7 8 9 10)(4 5 6)(2 3)(1)){e:('(.,.)
( : )' ' 
_'5 hrep)a e has#'  'hcat
}deepmap{e i:' 
 'i 3*hrep e,$hcat#/!LF+}map

Ungolfed

{ a :
  ((7 8 9 10) (4 5 6) (2 3) (1))
  { e :
    (
      '(.,.)' LF '( : )' + + 
      ' ' LF '_' + + 5 hrep
    ) @possible
    a e has @ind
    possible ind get @res
    '  ' @padding
    res padding hcat return
  } deepmap
  { e i:
    ' ' LF ' ' + + i 3 * hrep
    e ,
    $hcat insert!
    LF +
  } map
} @:bowl

(1 2 3 4 6 10) bowl out

Output:

(.,.) (.,.) (.,.)       
( : ) ( : ) ( : ) _____ 
         (.,.)       
   _____ ( : ) _____ 

      _____ _____ 

         _____ 

05AB1E, 45 44 bytes

TF"(.,.)( : )"„ _5×{«4ä2ä¹N>åè})4L£Rvyø»}».c

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Explanation

TF                                           # for N in [0 ... 9] do:
  "(.,.)( : )"                               # push string
              „ _                            # push the string " _"
                 5×                          # repeat it 5 times
                   {                         # sort
                    «                        # concatenate the strings
                     4ä                      # split the string in 4 parts
                       2ä                    # split the list in 2 parts
                         ¹N>åè               # if index+1 is in the input, push the first part
                                             # else push the second part
                              }              # end loop
                               )             # wrap stack in a list
                                4L£          # split list in parts of size 1,2,3,4
                                   R         # reverse list
                                    v        # for each list in list of lists
                                     yø      # transpose the list
                                       »     # join by spaces and newlines
                                        }    # end loop
                                         »   # join by newlines
                                          .c # centralize