Is it possible for a baby to be born a dwarf and also have down syndrome? How rare is a down dwarf?

Excedingly rare. I had occasion many years ago to investigate such a situation & needed to find answers for why some kids with ds chromosomes (t21) had totally different physical features. Run the numbers: achondroplasia (common new mutation dwarf) @ 1/15, 000 x 1/1700 (risk of ds @ 21)=1/25, 000, 000.