Traveling on a flat Earth

MATLAB, 327 304 296 290 282 276 267 259 255 253 bytes

I think I'm closing in on a limit now. I've managed to remove 51, 60 68 72 74 bytes from an answer I thought was well golfed. =)

Removed 8 bytes by taking the input inside the strsplit, thanks to Luis Mendo. Removed another 6 bytes by taking all the input into one cell array instead of two. Removed another 8 bytes by taking both last statements inside disp(). Removed 6 bytes by removing char and some brackets. This wasn't possible in a previous revision of the answer. Splitting the cell elements into four variables costed 15 bytes, but saved 24, thus 9 bytes saved! 8 additional bytes due to improved disp scheme. 5*~~s saves two bytes compared to (s>0)*5. Thus 4 new bytes saved (for s and t). Taking the opposite of this last expression saves yet another 2 bytes, 83-5*~s is shorter than 78+5*~~s

Golfed code:

x=strsplit([input('','s'),' ',input('','s')]);[a,b,c,d]=x{:};f=@str2num;n=@num2str;i=@sign;s=f(a(1:end-1))*i(a(end)-79)+f(c(1:end-1))*i(c(end)-79);t=f(b(1:end-1))*i(b(end)-79)+f(d(1:end-1))*i(d(end)-79);disp([n(abs(s)),83-5*~s,32,n(abs(t)),87-18*~t,''])

Same code, but with line breaks:

x=strsplit([input('','s'),' ',input('','s')]);
[a,b,c,d]=x{:};
f=@str2num;
n=@num2str;
i=@sign;
s=f(a(1:end-1))*i(a(end)-79)+f(c(1:end-1))*i(c(end)-79);
t=f(b(1:end-1))*i(b(end)-79)+f(d(1:end-1))*i(d(end)-79);
disp([n(abs(s)),78+5*~~s,32,n(abs(t)),69+18*~~t,''])

Test cases:

Save the above as flat_earth.m

flat_earth
0S 0E
0S 0W
0N 0E

flat_earth
0S 9999E
9999N 9999W
9999N 0E

flat_earth
42S 314W
42N 2718W
0N 3032W

flat_earth
5555N 8888W
7777S 0E
2222S 8888W

flat_earth
0001N        4545W
0999N        5454W
1000N 9999W

JavaScript (ES6), 118 bytes

s=>(g=([i,j,k,l,m,n],[o,p])=>(i=(j>p?-i:+i)+(n>p?-j:+j))<0?-i+o:i+p)(a=s.match(/\d+|\w/g),"SN")+' '+g(a.slice(2),"WE")

R, 196 bytes

R is pretty verbose, by golfing standards. Let's see...

i=scan(,'');l=nchar(i);S=W=`-`;N=E=c;n=as.double(substr(i,1,l-1));d=substr(i,l,l);for(j in 1:4)n[j]=get(d[j])(n[j]);o=c(n[1]+n[3],n[2]+n[4]);cat(paste0(abs(o), ifelse(o<0,c("S", "W"),c("N","E"))))

Ungolfed:

input = scan(,'')       # Take input from stdin, separating at newlines and spaces
length = nchar(input)   # Get the number of characters in each input
S=W=`-`                 # These two lines create aliases of `-` (the minus function)
N=E=c                   # and c (the concatenate function).
                        # We will later treat the NSEW part of the coordinate
                        # as a call to a function, to ensure that the numbers
                        # are treated with the correct polarity.
numbers = as.double(substr(input, 1, length-1))
                        # Strip the last character off of every coordinate, convert
                        # to integer
directions = substr(input, length, length)
                        # Strip off the numbers and grab the cardinal directions
for(j in 1:4)
    numbers[j] = get(directions[j])(numbers[j])
                        # For our four numbers, treat the cardinal direction as
                        # a function, which is mapped to `-` for south and west, and
                        # `c` for north and east (which is essentially identity)
output = c(numbers[1]+numbers[3], numbers[2]+numbers[4])
                        # Add together the numbers
cat(paste0(abs(output), ifelse(output<0, c("S", "W"), c("N","E"))))
                        # Output the absolute values of the new coordinates, followed
                        # by "S" or "W" if the number is negative and "N" or "E" if 
                        # the number is positive

Edit to add: I just looked at the other answers, and I'm surprised that my entry is one of the shortest! Perhaps R isn't as verbose as I thought...