Yes, three times in fact.
1994 beating duke and runner up the next year losing to UCLA
No, Illinois has never won the NCAA Men's Basketball Tournament.
No they have not.
Yes, they have won multiple titles, most famously under the coaching icon John Wooden: UCLA has a total of 11, which is the most out of any school in the NCAA's: 1964, 1965, 1967, 1968, 1969, 1970, 1971, 1972, 1973, 1975, 1995
no, they have not.
Twice. In 1980 and 1986.