do all double-reed and single-reed instruments belong to the woodwind family?

No.

Yes, both double-reed and single-reed instruments belong to the woodwind family of instruments. The woodwind family is characterized by the use of air blown through a reed or across a mouthpiece to produce sound. Double-reed instruments, such as the oboe and bassoon, use two small pieces of cane that vibrate when the player blows air between them. Single-reed instruments, such as the clarinet and saxophone, use a single reed attached to a mouthpiece. When air is blown through or across the reed, it causes the reed to vibrate and produce sound. These instruments are typically made of wood, hence the name "woodwind." However, there are also variants made of metal or other materials.