For question $1$, you can draw a Venn diagram for independent events, however you will not be able to tell if the events are independent by looking at the Venn diagram, as it will just look like a standard Venn diagram. (standard meaning how a Venn diagram would look for $2$ events $A,B$, that are not mutually exclusive)
For question $2$, if $A$, $B$ are $2$ independent events then $P(A\cup B)=P(A)+P(B)-P(A)P(B)$
For question $3$, a mutually exclusive events are necessarily dependent events (assuming the probability of both events is greater than $0$).
Proof:
Recall the following:
Let two events, $X$ and $Y$ be independent. Then it follows that $P(X \cap Y)=P(X)P(Y).$ These events are mutually exclusive if $P(X \cap Y)=0.$ Lastly remember that $P(X)>0$ and that also $P(Y)>0$, as we are discussing probability and it ranges from $0-1$.
So, since we know that $P(X)>0,P(Y)>0$, then it follows that $P(X)P(Y)>0.$ If these events were independent then $P(X)P(Y)=P(X\cap Y)>0$, but this would mean that they aren't mutually exclusive.
Therefore, the events can not be independent and mutually exclusive simultaneously if both their probabilities are more than $0$.
$Q.E.D.$
Best Answer
Your definition of independence, that $P(X|Y)=0$, is not the definition of independence used in probability. In fact, $P(X|Y)=0$ implies dependence, since the outcome of $Y$ influences (that is, completely inhibits) $X$.
What you're describing is mutual exclusivity.
When two outcomes are mutually exclusive, they are counted as two distinct possible outcomes of the same event. (e.g. a coin toss can be heads or tails. Heads or tails is two possible outcomes of one event: a single coin toss).
One the other hand, when we speak of two events being independent, we count these as two separate outcomes of two separate events (e.g. when tossing two coins, the outcome of either coin toss is independent of the other).