CJam, 46 bytes

Try it online, or try the test suite online.

lel{'n-_W>+_zE<*}%:+{_De<C~e>___0>-'n+'-?o-}h;

Explanation

The algorithm works how you might expect: read the input, convert to lowercase, map each character to a value, sum the values, and print characters and adjust the sum accordingly until the sum is zero. Probably the most interesting optimization (although it only saves two bytes) is that negated character mappings are used instead, as this avoids swapping subtraction arguments to correct the sign when computing the mapped value and avoids swapping again when mapping back to a letter due to the subtraction of a negated value being replaceable by addition.

lel             "Read a line of input and convert all letters to lowercase.";
{               "Map each character:";
  'n-_W>+         "Map each character to its negated value by subtracting 'n'
                   and incrementing if the result is nonnegative.";
  _zE<*           "If the value is out of the letter range, use 0 instead.";
}%
:+              "Compute the sum of the mapped character values.";
{               "Do...";
  _De<C~e>        "Compute the sum clamped to the letter range.";
  __              "If the clamped sum is nonzero, ...";
  _0>-'n+         "... then produce the clamped sum mapped back to a letter by
                     decrementing if it is positive and adding 'n', ...";
  '-              "... else produce '-'.";
  ?
  o               "Output the character produced above.";
  -               "Subtract the clamped sum out of the sum.";
}h              "... while the sum is nonzero.";
;               "Clean up.";

Pyth, 79 78 77 65 61 58

J+\-GK+0fTr13_14=ZsX@JzJKMe,pk*G/H13%H13@JxK?g\aZ>Z0_g\z_Z

Clip 10, 87

Fr+`m[y?cAyg#Ay-v,-RRZ]0]}m.U`[Fx?x?<x0,:-/xR'z*m'm%xR!!%xR],:/xR'a*m'n%xR!!%xR]]'-[R13

R, 258 bytes

function(s){a=13;r=data.frame;z=strsplit(gsub("[^a-z]","",tolower(s)),"")[[1]];d=r(l=rev(letters),h=c(-1*a:1,1:a));m=merge(r(l=z),d);n=sum(m$h);q=abs(n);v=rep(ifelse(n>0,"a","z"),q%/%a);paste0(paste(v,collapse=""),d$l[d$h==sign(n)*q%%a],ifelse(n==0,"-",""))}

This has to be the grossest R code ever. I figured R might be a decent choice since it has a vector of all letters "a" through "z" as a built-in global variable. But it turns out the rest is a mess.

Ungolfed + explanation:

function(s) {
    a <- 13              # Store the value associated with a
    r <- data.frame      # Store the `data.frame` function

    # Split the input into a vector, ignoring case and non-letters
    z <- strsplit(gsub("[^a-z]", "", tolower(s)), "")[[1]]

    # Create a data frame where the first column is the letters
    # z through a and the second is the associated scores
    d <- data.frame(l=reverse(letters), h=c(-1*a:1, 1:a))

    # Merge the split vector with the data frame of scores
    m <- merge(data.frame(l=z), d)

    # Get the total score for the input
    n <- sum(m$h)
    q <- abs(n)

    # Pad the output with a or z as necessary
    v <- rep(ifelse(n > 0, "a", "z"), q %/% a)

    # Collapse the vector of a's or z's into a string
    out1 <- paste(v, collapse="")

    # Look up the letter associated with remainder
    out2 <- d$l[d$h == sign(n)*q%%a]

    # If n = 0, we still want a dash
    out3 <- ifelse(n == 0, "-", "")

    # Return the concatenation of all pieces of the output
    paste0(out1, out2, out3)
}

This creates an unnamed function object that accepts a string as input and returns the associated hash value. To call it, give it a name, e.g. f=function(s){...}.

Examples:

> f("this is an example")
[1] "aak"

> f("root users")
[1] "zzs"

> f("love")
[1] "-"

> f("People like grapes.")
[1] "aaag"

Try it online!

Questions? I'll happily provide any further explanation. Suggestions? Suggestions are more than welcome!

Haskell, 171 bytes

import Data.Char
m=13
l=zip['a'..'z'][-i|i<-[-m..m],i/=0]
p 0="-"
p n|n>m='a':p(n-m)|n<(-m)='z':p(n+m)|1<2=[c|(c,i)<-l,i==n]
f n=p$sum[y|Just y<-[lookup(toLower x)l|x<-n]]

Test run:

> map f ["hello world", "love", "this is an example", "hello *&*(&(*&%& world", "good bye", "root users"]
["b","-","aak","b","ae","zzs"]

How it works: l is a lookup table from letters to the corresponding value. Lookup all characters from the input string and discard those not found. Sum the resulting list. Depending on the sum p prints - or maybe first some as or zs and finally (reverse-)lookups the letter from l.

R - 200

function(s){l=letters
N=setNames
x=strsplit(tolower(s),'')[[1]]
n=(13:-13)[-14]
v=sum(N(n,l)[x[x%in%l]])
o=''
while(v){d=min(max(v,-13),13)
o=paste0(o,N(l,n)[as.character(d)])
v=v-d}
if(o=='')o='-'
o}