Answers:
1. Quality control. Design was made by assuming material properties (such as strength) and that tests should be made to ensure that the assumption is correct.
2. Laboratory experiments gives scientific data under controlled conditions. Field experiments however is a far more effective approach on how the material is affected by the environment where it is intended to be used.
3. Tests shall be made before and after to see consistencies from fresh material (concrete) and the cured product. Sometimes, a material is prepared properly but doest not set well. This is the primary purpose of such tests.

I think, you are not really confused about life. With this question, i can see your ability to observe and analyse the events. Okay, now about your queries:
1. Let us take example of concrete. In all the countries standard codes are designed for selection of all engineering materials. Codes provide a standard procedure to test the materials. At laboratory we get exact result regarding compressive strength of concrete. As for example, we have designed the concrete mix targeted to achieve compressive strength of 35 Mpa, at laboratory we usually test on 30 samples and using compression testing machine or UTM we actually find the strength of samples prepared according to standards. Here if we do not get expected strength, we would bring changes in mix design so that our target strength is achieved.
Thus laboratory testing is necessery for validity of the mix prepared.
2. Now after getting confirmation from laboratory we start preparing mix at site. The whole required quantity of concrete can not be prepared at a time, so we do this in batches. Concrete is made of earthen material so chances of variation in quality of each batch are greater. Here, to make sure that each batch yields mix of same quality we need to employ any workability test, usually we prefer Slump test.
Thus, field tests are done to check quality control.
3. Now third thing you are talking about is Non destructive testing. After preparing and using concrete mix, we need to know whether it provides same compressive strength for which it was designed. Here after construction we dont have any other option excepting taking core samples and test them for compressive strength.
Thus, this is done for assuring the actual strength of concrete after construction. These tests must be employed at a prescribed regular interval of time for safety purpose.
Hope, i have made things clear and are useful to you.
Good Luck.