Yes, both in modern and traditional jazz.
is a cornet used in a jazz band
Clarinets are used in many different ensembles, rangng from Orchestras to swing bands. There are also clarinet choirs where all different types of clarinet play in harmony. Clarinets were also used alot in the Big Band era into the 1940s. Glann Millers Big Band found what it was looking for in the 1940s when a clarinettist joined them.
Usually a piano, bass, and drums, with one or two horns.
There is no "horn section" in a jazz band. Section implies a large jazz band and, as such, the sections would be referred to differently. "Horn" is an cover all word to describe all front line instruments, usually those which are mouth blown. There are, however, brass sections which are for trumpets, trombones and occasionally other brass instruments such as the euphonium and tuba, The woodwind section is for all the saxes, clarinets, and more rarely used instruments like the piccolo and oboe. Any reference, therefore, to a "horn section" is a misnomer.
There are many different instruments that can be used in a jazz band. The most common include saxophones, trumpets and horns, and trombones. Some jazz bands use other instruments such as flutes, drums, and vibraphones as well. There isn't an accurate explanation as to why because jazz bands are usually improvised and comprised of different people and different musical talents and sounds.
yes it can
Clarinets are used in any and every musical ensemble. Symphonic bands, symphonic orchestras, jazz, marching bands... Some places even have clarinet choirs devoted entirely to the various clarinets.
cello
That really depends on the sax. The Alto, Baritone, Tenor, and soprano saxes are all usually in a jazz band together but a soprano is not used in concert band. The alto,tenor,and bari are all in concert band.
Traditionally, clarinets are made from African hardwoods usually found in endangered rainforests. Consequently, the price of such wooden instruments was skyrocketing.
Big band and jazz
yes pianos were used in jazz music