What is Unique Coverage Problem?
Let's go over your questions one by one:

(1) Your definition of Unique cover is wrong. Given a set system $\mathcal{S}$, the goal is to find a subset $\mathcal{S}' \subseteq \mathcal{S}$ which uniquely covers the most elements. The subset $\mathcal{S}'$ uniquely covers an element $x$ if $x$ belongs to exactly one set in $\mathcal{S}'$.

(2) Exact cover is a decision problem, whereas Unique cover is an optimization problem. The decision version of Unique cover goes as follows:


  Given a set system and an integer $k$, is there a subset of the set system which uniquely covers at least $k$ elements?


Exact cover is a special case of the decision version of Unique cover.

(3) The sets $S_3$ and $S_4$ together cover $6$ elements once ($1,2,4,5,7,8$). Since $6 &gt; 4$, it is better to take both $S_3$ and $S_4$ rather than any of them individually.

(4) The probability is that of choosing any given set. I suggest you look for more material on probability theory and on randomized algorithms, and after that come back and see if it makes sense.

(5) This is explained very clearly in various lecture notes. There is no point repeating the explanation here. I suggest you learn probability theory and randomized algorithms from scratch, and then attempt to read the $O(\log n)$ analysis again. It might take a few weeks, but without catching up on the basics you have no chance of understanding anything which is even slightly advanced.