Bridge Hand Scoring

Jelly, 27 25 21 bytes

Thanks @Dennis for -3 bytes!

L_5.AḞW+“JQKA”i$€Sµ€S

This takes input as a list of lines. To take input as a multiline string, precatenate a ṣ⁷µ.

Making a table of the frequency points:

Cards:    0  1  2  3  4  5  6 ... 4+k
Points:   3  2  1  0  0  1  2 ... k

we can see that they're equal to abs(c-3.5)-.5, where c is the number of cards. Since each line contains two extra characters, and the number of points is always an integer, this is floor(abs(l-5.5)) where l is the line length.

Note that Jelly's indices are 1-based, and also the behavior of vectorized functions on mismatched dimensions: the extra elements of the longer list are unaffected. So [1] + [3,2,0,0] gives [4,2,0,0].

                  µ      The program is two monadic fs applied in turn; an atop.
L_5.AW+“JQKA”i$€S       Helper function:
                 €        For €ach line:
L                         Get the line Length.
 _5.                      Subtract 5.5 (Numeric literals' decimal parts default to .5)
    A                     Apply Absolute value
     Ḟ                    Floor
      W                   Then Wrap it in an array. "S:AKQT6" gives [1].
        “JQKA”i$          Monadic function: index into the string "JQKA".
                €         Apply ^ over €ach char of the line; [3,2,0,0,0].
       +                  Add the two arrays together; [4,2,0,0,0].
                 S        Sum; 6.
                    S    Main link: Sum all results

Try it here.

ES6, 107 99 89 bytes

s=>(t=0,[...s].map(c=>t+="JQKA".search(c)+1),s.split`
`.map(l=>t+=(l=l.length-6)^l>>4),t)

Pyth, 27 25 24 bytes

sms+a5.5ldshMxL"JQKA"d.z

We calculate the values separately for each suit, then add them.

  s m                 sum of map lambda d:  (d is a line of input)
      +                 add the
        s a                 floor of the absolute difference between
            5.5               5.5
            l d               and len(d)
          s hM xL           to the sum of the indices each incremented by one
                  "JQKA"      of each char in d in the string "JQKA"
                  d
      .z
   

Test suite.