Can I call a virtual method from a constructor/destructor?

Yes, but it's generally not a good idea. The mechanics of object construction in .NET are quite different from C++, and this affects virtual method calls in constructors.
C++ constructs objects from base to derived, so when the base constructor is executing the object is effectively a base object, and virtual method calls are routed to the base class implementation. By contrast, in .NET the derived constructor is executed first, which means the object is always a derived object and virtual method calls are always routed to the derived implementation. (Note that the C# compiler inserts a call to the base class constructor at the start of the derived constructor, thus preserving standard OO semantics by creating the illusion that the base constructor is executed first.)
The same issue arises when calling virtual methods from C# destructors. A virtual method call in a base destructor will be routed to the derived implementation.

How do I use the 'using' keyword with multiple objects?

You can nest using statements, like this:
using( obj1 )
{
using( obj2 )
{
...
}
}
However consider using this more aesthetically pleasing (but functionally identical) formatting:
using( obj1 )
using( obj2 )
{
...
}

What is the difference between == and object.Equals?

For value types, == and Equals() usually compare two objects by value. For example:
int x = 10;
int y = 10;
Console.WriteLine( x == y );
Console.WriteLine( x.Equals(y) );
will display:
True
True
However things are more complex for reference types. Generally speaking, for reference types == is expected to perform an identity comparison, i.e. it will only return true if both references point to the same object. By contrast, Equals() is expected to perform a value comparison, i.e. it will return true if the references point to objects that are equivalent. For example:
StringBuilder s1 = new StringBuilder("fred");
StringBuilder s2 = new StringBuilder("fred");
Console.WriteLine( s1 == s2 );
Console.WriteLine( s1.Equals(s2) );
will display:
False
True
s1 and s2 are different objects (hence == returns false), but they are equivalent (hence Equals() returns true).
Unfortunately there are exceptions to these rules. The implementation of Equals() in System.Object (the one you'll inherit by default if you write a class) compares identity, i.e. it's the same as operator==. So Equals() only tests for equivalence if the class author overrides the method (and implements it correctly). Another exception is the string class - its operator== compares value rather than identity.
Bottom line: If you want to perform an identity comparison use the ReferenceEquals() method. If you want to perform a value comparison, use Equals() but be aware that it will only work if the type has overridden the default implementation. Avoid operator== with reference types (except perhaps strings), as it's simply too ambiguous.

How do I enforce const correctness in C#?

You can't - at least not in the same way you do in C++. C# (actually, the CLI) has no real concept of const correctness, For example, there's no way to specify that a method should not modify an argument passed in to it. And there's no way to specify that a method does not modify the object on which it is acting.
To get a feel for the angst this causes among some C++ programmers, read the feedback on this post from Raymond Chen.
There are of course ways of addressing this issue. For example, see Brad Abram's post (and associated feedback) for some ideas on adding optional read-only behaviour to collection classes.