Does Thread lock on object ensures lock on member objects too?

Consider following class that I have wrote for testing the locking of non-primitive variable (myObject). My question is:

(Consider all threads are working on same object instance of SynchronizationTest)

I understand that if thread1 is executing set(...) method then any other thread (lets say thread2) is okay to execute either of the anotherSetWithSynchronized(...) or anotherSetWithoutSynchronized(...).

Question a : If thread1 has locked the object of SynchronizationTest while executing set(...), does it mean it has acquired lock on all member object variable ? like in this case myObject. If not then,

Question b: If thread1 is executing set(...) can thread2 execute anotherSetWithSynchronized(...) concurrently ?

Ouestion C: None of the two methods can execute simultaneously by thread?

Ouestion D: Design is wrong, I need to explicitly lock myObject in synchronized set(...) method. Like this:

  public synchronized void set(MyValue myValue)
   {
     synchronized (myObject)
     {
       myObject.put(myValue);
     }
   } 
 

======================SynchronizationTest class======================

public class SynchronizationTest
 {
   private MyObject myObject = new MyObject();
 
   public synchronized void set(MyValue myValue)
   {
     myObject.put(myValue);
   }
 
   public void anotherSetWithSynchronized(MyValue myValue)
   {
     synchronized (myObject)
     {
       myObject.put(myValue);
     }
   }
 
   public void anotherSetWithoutSynchronized(MyValue myValue)
   {
     myObject.put(myValue);
   }
 }
 

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