C, 88 87 86 81 bytes

c,d;i(char*a,char*b){for(c=d=0;*b;c++,a++)d+=*a^*b++?*a?-c/15-c/20:0:5+c/15;d=d;}

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Jelly, 26 23 22 bytes

=s5ị"“HHHQP‘D¤_2Fæ.n⁶¥

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How it works

=s5ị"“HHHQP‘D¤_2Fæ.n⁶¥  Main link. Argument: t (answer to test), s (answer sheet)

=                       Test the characters of t and s for equality.
 s5                     Split into chunks of length 5.
             ¤          Combine the two preceding links into a niladic chain.
     “HHHQP‘              Yield the code points, i.e., [72, 72, 72, 81, 80].
            D             Decimal; yield [[7, 2], [7, 2], [7, 2], [8, 1], [8, 0]].
   ị"                   Index zipwith; use the Booleans in each chunk to index into
                        the corresponding pair. Indexing is 1-based and modular, so
                        1 gives the first element and 0 the last.
              _2        Subtract 2 from each result.
                F       Flatten the resulting 5x5 matrix.
                     ¥  Combine the two preceding links into a dyadic chain.
                   n⁶     Test the characters of t for inequality with space.
                 æ.     Take the dot product of the integers to the left and the
                        Booleans to the right.

JavaScript (ES6), 70 68 66 bytes

Saved 2 bytes thanks to Neil
Saved 2 bytes thanks to ETHproductions

Takes applicant answers a and correct answers c in currying syntax (a)(c). Expects skipped questions to be marked with a space.

a=>c=>a.replace(/\S/g,(a,i)=>s+=a==c[j=i>14,i]?5+j:-j^i>19,s=0)&&s

Test cases

let f =

a=>c=>a.replace(/\S/g,(a,i)=>s+=a==c[j=i>14,i]?5+j:-j^i>19,s=0)&&s

console.log(f
  ("DDDDDDDDDDDDDDDDDDDDDDDDD")
  ("ABCDEABCDEABCDEABCDEABCDE")
);
console.log(f
  ("BDBEACCECEDDBDABBCBDAEBCD")
  ("BDBEACCECEDDBDABBCBDAEBCD")
);
console.log(f
  ("DBACBDCDBAEDABCDBEECACDC ")
  ("DBADBDCDBAEDABCDBEEDACDCA")
);
console.log(f
  ("                         ")
  ("DABDABDABDABDABDABDABDABD")
);
console.log(f
  ("DBADBDCDBAEDABCD E  A  C ")
  ("DBADBDCDBAEDABCDBEEDACDCA")
);
console.log(f
  ("               BBBBBBBBBB")
  ("AAAAAAAAAAAAAAAAAAAAAAAAA")
);

Mathematica, 114 bytes

Tr@(m=MapThread)[#/.True->#2/.False->-#3&,{Tr/@Partition[m[Equal,#/."_"->u]/.u==_->0,5],{5,5,5,6,6},{0,0,0,1,2}}]&

Pure function taking an ordered pair of lists of characters and returning an integer. m[Equal,#/."_"->u] returns a list of booleans, except for unevaluated entries of the form u=="B" in places where the answer equaled "_"; then right away, u==_->0 turns those unevaluated entries into 0s. Tr/@Partition[...,5] adds these entries up 5 at a time, resulting in a list like {4False+True, 4False+True, 4False+True, 4False+True, 4False+True} for the first test case or {5True, 5True, 5True, 2True, 2True} for the last test case. Then in each coordinate, True and False are mapped to the appropriate scores, and the results are added together.

Python 2, 93 91 bytes

f=lambda a,b,n=0:a>""and((a[0]==b[0])*(5+n/15)or-(n/15*n/10)*(a[0]<"^"))+f(a[1:],b[1:],n+1)

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-2 bytes thanks to @KritixiLithos

Input:

• a : Answers of the student as a string, _ for skipped question
• b : correct answers
• n : the number of the current question 0-based, defaults to 0

k, 52 bytes

Function takes 2 strings, format per the test cases

{+/(~x="_")*(+,/'(15 5 5#'0 -1 -2;15 10#'5 6))@'x=y}

Example:

k){+/(~x="_")*(+,/'(15 5 5#'0 -1 -2;15 10#'5 6))@'x=y}["DBADBDCDBAEDABCD_E__A__C_";"DBADBDCDBAEDABCDBEEDACDCA"]
99

Haskell, 84 bytes

i x a b|a>'Z'=0|a==b=6-0^x|1<2= -x
w x=x<$[1..5*3^0^x]
(sum.).zipWith3 i(w=<<[0..2])

Usage example: ((sum.).zipWith3 i(w=<<[0..2])) "DBADBDCDBAEDABCD_E__A__C_" "DBADBDCDBAEDABCDBEEDACDCA" -> 99. Try it online!.

How it works: i x a b calculates the score for a single answer a with correct result b and the penalty x for a wrong answer (a non-negative value). If you skip (a>'Z'), the score is 0, if the answer is right (a==b), the score is 6-0^x, else the score is -x.

w=<<[0..2] makes a list of penalties for all 25 questions by applying w to 0, 1 and 2, i.e. making 5*3^0^x copies of each number (-> 15 times 0, 5 times 1 and 5 times 2).

zipWith3 applies i to the list of penalties, list of answers and list of correct results. Finally all scores are added (sum).

JavaScript (ES6), 105 103 101 94 89 88 85 84 78 bytes

My first solution in ES6, maybe even first in Javascript O.o

s=>a=>eval([...s].map((c,i)=>c>'Z'?0:c==a[i]?5+(i>14):~(i>19)*(i>14)).join`+`)

Takes input with a currying syntax (s)(a), where s is the submitted solution and a is the correct solution. Both will be taken as strings.

Thanks to @ETHproductions for saving 9 bytes! s[i] to c and (-1-(i>19|0)) to ~(i>19).

Thanks to @Kritixi Lithos for saving a byte! c=='_' to c>'Z'.

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Python 2, 86 85 83 77 bytes

f=lambda t,s,i=24:~i and(i/10*-(14<i<t[i]<'_'),5+i/15)[t[i]==s[i]]+f(t,s,i-1)

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How it works

This defines a recursive function f that takes two non-optimal arguments: t (the answers to the test) and s (the answer sheet). When called only with these two arguments, f initializes i to 24, the last index of both t and s.

Every time f is called, it first checks if ~i (the bitwise NOT of i) is truthy/non-zero. Since ~(-1) = 0, this happens once the i reaches the value -1. If i = -1, ~i = 0 is returned, but as i takes values from 24 to 0 (all indices of t and s), the code following and is executed and f returns the result.

While i is non-negative, the following happens. First,

(i/10*-(14<i<t[i]<'_'),5+i/15)

creates a tuple of length 2:

• The quotient i/10 is 0 if 0 ≤ i < 10, 1 if 10 ≤ i < 20, and 2 if 20 ≤ i < 25. The chained comparison 14<i<t[i]<'_' returns True if and only if all individual comparisons return True, i.e., if and only if i ≥ 15 (the range of questions with penalty), i is smaller than t[i] (always true since all numbers are smaller than all iterables in Python 2), and t[i] is not an underscore.

  If the comparison returns False, the unary - returns 0 and the entire expression evaluates to 0. However, if the comparison returns True, the unary - returns -1, so the entire expression evaluates to 0 if 0 ≤ i < 10, -1 if 10 ≤ i < 20, and -2 if 20 ≤ i < 25; these are the net results for wrong or missing answers for all indices i.

• 5+i/15 returns 5 + 0 = 5 if 0 ≤ i < 15 and 5 + 1 = 6 if 15 ≤ i < 25. These are the net results for correct answers for all indices i.

Finally, [t[i]==s[i]] selects the first element of the constructed tuple if t[i] and s[i] differ (wrong or missing answer) and the second one if they are equal (correct answer), then adds the return value of f called with decremented i to that result. Once i reaches -1, the final score has been computed and is returned by f.

Octave, 61 54 bytes

@(a,b)[a==b,-(a<95&a~=b)]*[(x=1:25>15)+5,(1:25>20)+x]'

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Previous answer:

@(a,b)(m=a==b)*(((f=kron(z=[0 0 0:2],z|1)')&1)+5)-(a<95&~m)*f