XML is a subset of SGML. A valid XML document is also a valid SGML document, but the vice versa isn't true.
Does this keep true even if I use the XML Schema instead of the DTD?
The doubt stems from that fact that the XML DTD is a subset of the SGML DTD, but the XML Schema allows things that can't be defined in the XML DTD.
So is XML Schema still a subset? Can I do with SGML DTD everything I can do with XML Schema?
Is XML Schema a subset of SGML DTD?
No, XML Schema is far from being a subset of DTD. XML Schema uses a completely different syntax than DTD, and XML Schema can express many constraints not possible in DTD.
A valid XML document is also a valid SGML document
You probably mean well-formed, not valid. Well-formed is not the same as valid. Here is probably where you want to make a subset statement. XML was designed to be a subset of SGML, yes.
Can I do with SGML DTD everything I can do with XML Schema?
No, not even close. For just one example, DTDs cannot restrict data to numeric types, but XML Schema can trivially. XML Schema in general is far more expressive than DTD.