Counting max and min per row across columns and outputting associated column names

With your shown samples, please try following awk code. Written and tested in GNU awk.

awk -v startField="3" -v endField="6" '
BEGIN{ OFS="\t"; print "chr             pos     ref     max     ref     min"}
FNR==1{
  for(i=startField;i<=endField;i++){
    heading[i]=$i
  }
  next
}
{
  min=max2=maxInd2=minInd=max=maxInd=minAllInd=maxAllInd=maxAllInd2=""
  for(i=startField;i<=endField;i++){
    if($i!=0){
      minInd=(min>$i?i:(min==$i?minInd","i:(minInd!=""?minInd:i)))
      min=(min>$i?$i:(min!=""?min:$i))
    }
    maxInd=(max<$i?i:(max==$i?maxInd","i:(maxInd!=""?maxInd:i)))
    max=(max<$i?$i:(max!=""?max:$i))
  }
  for(i=startField+1;i<=endField;i++){
    maxInd2=(max2<$i?i:(max2==$i?maxInd2","i:(maxInd2!=""?maxInd2:i)))
    max2=(max2<$i?$i:(max2!=""?max2:$i))
  }
  num1=split(maxInd,arr1,",")
  num2=split(minInd,arr2,",")
  num3=split(maxInd2,arr3,",")
  if(num1>1){
    for(k=1;k<=num1;k++){
       maxAllInd = (maxAllInd?maxAllInd ",":"") heading[arr1[k]]
    }
  }
  else{
    maxAllInd = heading[maxInd]
  }
  if(num2>1){
    for(k=1;k<=num2;k++){
       minAllInd = (minAllInd?minAllInd ",":"") heading[arr2[k]]
    }
  }
  else{
    minAllInd = heading[minInd]
  }
  if(num3>1){
    for(k=1;k<=num3;k++){
       maxAllInd2 = (maxAllInd2?maxAllInd2 ",":"") heading[arr3[k]]
    }
  }
  else{
    maxAllInd2 = heading[maxInd2]
  }
  if(startField>1){
    NF=(startField-1)
      if(min !=0 ){
         print $0,maxAllInd,max,minAllInd,min
      }
      if(min == 0 && max2 != 0){
         print $0,maxAllInd,max,maxAllInd2,max2
      }
      if(min == 0 && max2 == 0){
         print $0,maxAllInd,max,maxAllInd,max
      }  
  }
  else{
      if(min !=0 ){
         print maxAllInd,max,minAllInd,min
      }
      if(min == 0 && max2 != 0){
         print maxAllInd,max,maxAllInd2,max2
      }
      if(min == 0 && max2 == 0){
         print maxAllInd,max,maxAllInd,max
      }       
  }
}
'  Input_file

• You have some typos in variable names such as basemmin and mim.
• If the count of C is 0, the min value has no chance to be updated.
• You can combine the two for loops into one.
• The variable l is not used.

Then would you please try the following:

awk -v OFS="\t" '
NR==1 {
    print $1, $2, "ref", "max", "ref", "min"
    for (i = 3; i <= 6; i++) bases[i] = $i
}
NR>1 {
    basemax = bases[3]; basemin = bases[3]; max = $3; min = $3
    for (i = 4; i <= 6; i++) {
        if ($i > max) {basemax = bases[i]; max = $i}
        else if ($i == max) {basemax = basemax "," bases[i]}
        if ($i < min && $i != 0 || min == 0) {basemin = bases[i]; min = $i}
        else if ($i == min) {basemin = basemin "," bases[i]}
    }
    print $1, $2, basemax, max, basemin, min
}' input_file

Output:

chr     pos     ref     max     ref     min
NC_044998.1     3732    C       22      A       7
NC_044998.1     3733    C       22      C       22
NC_044998.1     3734    C       22      T,A     3
NC_044998.1     3735    C       22      G       3
NC_044998.1     3736    A       22      G       3
NC_044998.1     3737    G       25      G       25
NC_044998.1     3738    C       22      T       7
NC_044998.1     3739    G       25      T       3
NC_044998.1     3740    T,A     22      T,A     22
NC_044998.1     3741    C       22      C       22

Please note the output slightly differs from your desired output, which may contain typos.

This awk script should work for you:

cat maxmin.awk

NR == 1 {
   for (i=b; i<=NF; ++i)
      hdr[i] = $i
   print $1, $2, "ref", "max", "ref", "min"
   next
}
{
   for (i=b; i<=NF; ++i) {
      max = ($i > max ? $i : max)
      min = ($i && (min == "" || $i < min) ? $i : min)
   }
   for (i=b; i<=NF; ++i) {
      if ($i == min)
         rmin = (rmin ? rmin "," : "") hdr[i]
      if ($i == max)
         rmax = (rmax ? rmax "," : "") hdr[i]
   }
   print $1, $2, rmax, max, rmin, min
   max = min = rmax = rmin = ""
}

And use it as:

awk -v b=3 -f maxmin.awk gg | column -t
chr          pos   ref  max  ref  min
NC_044998.1  3732  C    22   A    7
NC_044998.1  3733  C    22   C    22
NC_044998.1  3734  C    22   T,A  3
NC_044998.1  3735  C    22   G    3
NC_044998.1  3736  A    22   G    3
NC_044998.1  3737  G    25   G    25
NC_044998.1  3738  C    22   T    7
NC_044998.1  3739  G    25   T    3
NC_044998.1  3740  T,A  22   T,A  22
NC_044998.1  3741  C    22   C    22

column -t has been used for tabular output only.