The quads
Both. Concentric: Exercise where the muscle is contracted under a load Eccentric: Exercise where the muscle is stretched under a load. So when you squat down its eccentric, when you come back up its concentric. Most exercises are both, with a few exceptions (i.e. jumping of a wall is mainly eccentric, while a punch is mainly concentric. Eccentric contraction is associated with increased muscle growth, but also DOMS (Delayed onset muscle soreness, when your muscles are very sore after a workout).
isometric
Rapid stretching and contracting of muscles. Ex) squat jump, repeat squat jump.
Glutes and quads contract when you stand, lengthen when you squat. So it's eccentric for those muscles. There are a bunch of muscles involved in a squat, some may be concentric.
Squat.
Quad
mainly your legs but and your back a little
Yes, it is a secondary muscle (synergist/stabilizer) used in the squat. There is a link below in which you can get a chart of the muscles used in a squat: go down about halfway and you'll see the chart.
Body weight squat, goblet squat, zombie squat, front squat, and back squat
Box Squat is a variation of a squat where you sit on a box and squat back up.
i dont know all of them, but your calves will be limp noodles the day after you do a bunch. thighs and butt, probably, too.
The squat works many muscles in the body in some form. The primary movers are the hamstrings, quads, and glutes. Assuming you are doing them correctly with your knees out and squatting below parallel, you should be using your hip adductors as well.