Let's take one at a time:
Yes, a diester is a molecule containing two ester groups. For example, animal fat is a triester.
If you add silver nitrate solution, you are actually adding Ag
+(aq) ions. These form a cream ppt with any bromide ions present. Tertiary bromo compounds actually exist in equilibrium with the corresponding carbocation and bromide ions, so you could assume that there are some present.
You are correct that both K and M give positive tests with carbonate ions. However, the question says minimum number of tests. That implies that you may need more than one.
For the DNA you have to learn the fundamental structure:
DNA10.4. There are three unique environments for the carbon atoms and therefore three signals. It does not say triplet, it says three signals.