You can invoke private method with reflection. Modifying the last bit of the posted code:

Method method = object.getClass().getDeclaredMethod(methodName);
method.setAccessible(true);
Object r = method.invoke(object);

There are a couple of caveats. First, getDeclaredMethod will only find method declared in the current Class, not inherited from supertypes. So, traverse up the concrete class hierarchy if necessary. Second, a SecurityManager can prevent use of the setAccessible method. So, it may need to run as a PrivilegedAction (using AccessController or Subject).

1) How can I access the private methods and the private data members?

You can do it with a little help of the setAccessible(true) method:

class Dummy{
    private void foo(){
        System.out.println("hello foo()");
    }
    private int i = 10;
}

class Test{
    public static void main(String[] args) throws Exception {
        Dummy d = new Dummy();

        /*---  [INVOKING PRIVATE METHOD]  ---*/
        Method m = Dummy.class.getDeclaredMethod("foo");
        //m.invoke(d); // Exception java.lang.IllegalAccessException
        m.setAccessible(true);//Abracadabra
        m.invoke(d); // Now it's OK

        /*---  [GETING VALUE FROM PRIVATE FIELD]  ---*/
        Field f = Dummy.class.getDeclaredField("i");
        //System.out.println(f.get(d)); // Not accessible now
        f.setAccessible(true); // Abracadabra
        System.out.println(f.get(d)); // Now it's OK

        /*---  [SETTING VALUE OF PRIVATE FIELD]  ---*/
        Field f2 = Dummy.class.getDeclaredField("i");
        //f2.set(d,20); // Not accessible now
        f2.setAccessible(true); // Abracadabra
        f2.set(d, 20); // Now it's OK
        System.out.println(f2.get(d));
    }
}

2) Is it possible to access a local variable via reflection?

No. Local variables cannot be accessed outside of a block in which they were created (someone could say that you can assign such a variable to a field like field = localVariable; and later access such a field via reflection, but this way we will be accessing the value, not the variable).

3) Is there any way to prevent anyone from accessing private constructors, methods, and data members?

I think for constructors or methods you could use stacktrace to check if it was invoked by Reflection.
For fields I can't find a solution to prevent accessing them via reflection.

[WARNING: This is not approved by anyone. I just wrote it inspired by your question.]

class Dummy {
    private void safeMethod() {
        StackTraceElement[] st = new Exception().getStackTrace();
        // If a method was invoked by reflection, the stack trace would be similar
        // to something like this:
        /*
        java.lang.Exception
            at package1.b.Dummy.safeMethod(SomeClass.java:38)
            at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
            at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
            at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
        ->    at java.lang.reflect.Method.invoke(Method.java:601)
            at package1.b.Test.main(SomeClass.java:65)
        */
        //5th line marked by "->" is interesting one so I will try to use that info

        if (st.length > 5 &&
            st[4].getClassName().equals("java.lang.reflect.Method"))
            throw new RuntimeException("safeMethod() is accessible only by Dummy object");

        // Now normal code of method
        System.out.println("code of safe method");
    }

    // I will check if it is possible to normally use that method inside this class
    public void trySafeMethod(){
        safeMethod();
    }

    Dummy() {
        safeMethod();
    }
}

class Dummy1 extends Dummy {}

class Test {
    public static void main(String[] args) throws Exception {
        Dummy1 d1 = new Dummy1(); // safeMethod can be invoked inside a superclass constructor
        d1.trySafeMethod(); // safeMethod can be invoked inside other Dummy class methods
        System.out.println("-------------------");

        // Let's check if it is possible to invoke it via reflection
        Method m2 = Dummy.class.getDeclaredMethod("safeMethod");
        // m.invoke(d);//exception java.lang.IllegalAccessException
        m2.setAccessible(true);
        m2.invoke(d1);
    }
}

Output from Test main method:

code of safe method
code of safe method
-------------------
Exception in thread "main" java.lang.reflect.InvocationTargetException
    at sun.reflect.NativeMethodAccessorImpl.invoke0(Native Method)
    at sun.reflect.NativeMethodAccessorImpl.invoke(NativeMethodAccessorImpl.java:57)
    at sun.reflect.DelegatingMethodAccessorImpl.invoke(DelegatingMethodAccessorImpl.java:43)
    at java.lang.reflect.Method.invoke(Method.java:601)
    at package1.b.Test.main(MyClass2.java:87)
Caused by: java.lang.RuntimeException: method safeMethod() is accessible only by Dummy object
    at package1.b.Dummy.safeMethod(MyClass2.java:54)
    ... 5 more

use setAccessible(true) on your Method object before using its invoke method.

import java.lang.reflect.*;
class Dummy{
    private void foo(){
        System.out.println("hello foo()");
    }
}

class Test{
    public static void main(String[] args) throws Exception {
        Dummy d = new Dummy();
        Method m = Dummy.class.getDeclaredMethod("foo");
        //m.invoke(d);// throws java.lang.IllegalAccessException
        m.setAccessible(true);// Abracadabra 
        m.invoke(d);// now its OK
    }
}

If someone is interested in invoking methods with parameters see How do I invoke a Java method when given the method name as a string?, specifically answer like https://stackoverflow.com/a/30671481.

Just don't forget to add setAccessible(true) while invoking private methods.