UPDATE: With Spring 4.1 it is possible to set the user on the handshake for #1 from above.  Per the Spring documentation you can create a new class which extends DefaultHandshakeHandler and override the determineUser method.  Additionally you can also create a security filter which sets the principal as well if you have a token.  I have implemented the second one myself and I include some sample code for both below.
For #2 and #3 I do not think that it is possible still.  For #4 Spring intentionally ignores these per the documentation here.
SAMPLE CODE FOR DefaultHandshakeHandler SUBCLASS:
@Configuration
@EnableWebSocketMessageBroker
public class ApplicationWebSocketConfiguration extends AbstractWebSocketMessageBrokerConfigurer {
    public class MyHandshakeHandler extends DefaultHandshakeHandler {
        @Override
        protected Principal determineUser(ServerHttpRequest request, WebSocketHandler wsHandler, 
                                          Map<String, Object> attributes) {
            // add your own code to determine the user
            return null;
        }
    }
    @Override
    public void registerStompEndpoints(StompEndpointRegistry registry) {
        registry.addEndpoint("/myEndPoint").setHandshakeHandler(new MyHandshakeHandler());
    }
}
SAMPLE CODE FOR SECURITY FILTER:
public class ApplicationSecurityTokenFilter extends GenericFilterBean {
    private final static String AUTHENTICATION_PARAMETER = "authentication";
    @Override
    public void doFilter(ServletRequest servletRequest, ServletResponse servletResponse, FilterChain filterChain) throws IOException, ServletException {
        if (servletRequest instanceof HttpServletRequest) {
            // check to see if already authenticated before trying again
            Authentication existingAuth = SecurityContextHolder.getContext().getAuthentication();
            if ((existingAuth == null) || !existingAuth.isAuthenticated()) {
                HttpServletRequest request = (HttpServletRequest)servletRequest;
                UsernamePasswordAuthenticationToken token = extractToken(request);
                // dump token into security context (for authentication-provider to pick up)
                if (token != null) {  // if it exists
                    SecurityContextHolder.getContext().setAuthentication(token);
                }
            }
        }
        filterChain.doFilter(servletRequest,servletResponse);
    }
    private UsernamePasswordAuthenticationToken extractToken( HttpServletRequest request ) {
        UsernamePasswordAuthenticationToken authenticationToken = null;
        // do what you need to extract the information for a token
        // in this example we assume a query string that has an authenticate
        // parameter with a "user:password" string.  A new UsernamePasswordAuthenticationToken
        // is created and then normal authentication happens using this info.
        // This is just a sample and I am sure there are more secure ways to do this.
        if (request.getQueryString() != null) {
            String[] pairs = request.getQueryString().split("&");
            for (String pair : pairs) {
                String[] pairTokens = pair.split("=");
                if (pairTokens.length == 2) {
                    if (AUTHENTICATION_PARAMETER.equals(pairTokens[0])) {
                        String[] tokens = pairTokens[1].split(":");
                        if (tokens.length == 2) {
                            log.debug("Using credentials: " + pairTokens[1]);
                            authenticationToken = new UsernamePasswordAuthenticationToken(tokens[0], tokens[1]);
                        }
                    }
                }
            }
        }
        return authenticationToken;
    }
}
// set up your web security for the area in question
@Configuration
public class SubscriptionWebSecurityConfigurationAdapter extends WebSecurityConfigurerAdapter {
    protected void configure(HttpSecurity http) throws Exception {
        http
                .requestMatchers().antMatchers("/myEndPoint**","/myEndPoint/**").and()
                .addFilterBefore(new ApplicationSecurityTokenFilter(), UsernamePasswordAuthenticationFilter.class)
                .authorizeRequests()
                .anyRequest().authenticated()
                .and()
                .httpBasic()  // leave this if you want non web browser clients to connect and add an auth header
                .and()
                .csrf().disable();
    }
}
** NOTE: ** DO NOT declare your filter as a Bean.  If you do then it will also be picked up (at least using Spring Boot) in the generic filters so it will fire on every request.