Q1: When drainage is permitted under initially applied normal stress only and full primarily consolidation is allowed to take place, the test is known as

ANS:C - consolidated undrained test

Quick test is the test where drainage is not allowed i.e. unconsolidated undrained test.
The drained test is the test in which water is allowed to drain after 2nd pressure. Drainage allowed in 1st stage and not in 2nd stage = consolidated undrained stage.
Drainage is allowed in both 1st and 2nd stages = consolidated drained stage.
Drainage is not allowed in both 1st and 2nd stages = the unconsolidated undrained stage. The unconsolidated undrained test is called a quick test.
So, a quick test is different from the consolidated undrained test I think, seepage means allowing the water through the soil pores.
Consolidation means expulsion (removing) of water voids from soil by applying some static loads . When drainage is permitted under initial applied stress that means we are already consolidated and applied stress without permitted water i.e. consolidated undrained test.