How do I find out the browser’s proxy settings?

The function you’re looking for is WinHttpGetIEProxyConfigForCurrentUser(), which is documented at http://msdn.microsoft.com/en-us/library/aa384096(VS.85).aspx. This function is used by Firefox and Opera to get their proxy settings by default, although you can override them per-browser. Don’t do that, though. The right thing to do (which is what everybody else does) is to just get the IE settings and assume that they’re correct, since they almost always are.

Here’s a sample of the relevant logic, which you should adapt for your needs:

if( WinHttpGetIEProxyConfigForCurrentUser( &ieProxyConfig ) )
{
    if( ieProxyConfig.fAutoDetect )
    {
        fAutoProxy = TRUE;
    }

    if( ieProxyConfig.lpszAutoConfigUrl != NULL )
    {
        fAutoProxy = TRUE;
        autoProxyOptions.lpszAutoConfigUrl = ieProxyConfig.lpszAutoConfigUrl;
    }
}
else
{
    // use autoproxy
    fAutoProxy = TRUE;
}

if( fAutoProxy )
{
    if ( autoProxyOptions.lpszAutoConfigUrl != NULL )
    {
        autoProxyOptions.dwFlags = WINHTTP_AUTOPROXY_CONFIG_URL;
    }
    else
    {
        autoProxyOptions.dwFlags = WINHTTP_AUTOPROXY_AUTO_DETECT;
        autoProxyOptions.dwAutoDetectFlags = WINHTTP_AUTO_DETECT_TYPE_DHCP | WINHTTP_AUTO_DETECT_TYPE_DNS_A;
    }

    // basic flags you almost always want
    autoProxyOptions.fAutoLogonIfChallenged = TRUE;

    // here we reset fAutoProxy in case an auto-proxy isn't actually
    // configured for this url
    fAutoProxy = WinHttpGetProxyForUrl( hiOpen, pwszUrl, &autoProxyOptions, &autoProxyInfo );
}

if ( fAutoProxy )
{
    // set proxy options for libcurl based on autoProxyInfo
}
else
{
    if( ieProxyConfig.lpszProxy != NULL )
    {
        // IE has an explicit proxy. set proxy options for libcurl here
        // based on ieProxyConfig
        //
        // note that sometimes IE gives just a single or double colon
        // for proxy or bypass list, which means "no proxy"
    }
    else
    {
        // there is no auto proxy and no manually configured proxy
    }
}