How can I hash a string to an int using c++?

From personal experience I know that this works and produces good distributions. (Plagiarised from http://www.cse.yorku.ca/~oz/hash.html):

djb2

this algorithm (k=33) was first reported by dan bernstein many years ago in comp.lang.c. another version of this algorithm (now favored by bernstein) uses xor: hash(i) = hash(i - 1) * 33 ^ str[i]; the magic of number 33 (why it works better than many other constants, prime or not) has never been adequately explained.

unsigned long hash(unsigned char *str) {
    unsigned long hash = 5381;
    int c;

    while (c = *str++) {
        hash = ((hash << 5) + hash) + c; /* hash * 33 + c */
    }

    return hash;
}

Re the first question, sure, e.g, something like:

int hash = 0;
int offset = 'a' - 1;
for(string::const_iterator it=s.begin(); it!=s.end(); ++it) {
  hash = hash << 1 | (*it - offset);
}

regarding the second, there are many better ways to hash strings. E.g., see here for a few C examples (easily translatable to C++ along the lines of the snippet above).

You can examine each individual char from a std::string using the [] operator. However, you can look at Boost::Functional/Hash for guidance on a better hashing scheme. There is also a list of hashing functions in c located here.