Let's start with Question B:
B. Does this (necessarily) mean the first attempt was successful?
Yes, it does. The expression x for y (where x and y are generally relatively small integers), means x successes, out of y attempts. So:
He was two for three against Boston
means he had two successes in three attempts. (The meaning is context dependent: in basketball, we might be talking about free throws; in baseball, we might be talking about at bats. If Boston is the last name of a politician, we could be talking about the results of three mayoral elections, where Mr. Boston won once.)
So, in the case of:
Let's see how my luck goes – shall I try two for two?
If the first attempt was unsuccessful, the best the speaker could try for is one for two. Hence, the first attempt must have been successful. The speaker is now 1 for 1, not 0 for 1 ("0 for 1" is usually pronounced as "oh for one," at least in the U.S.).
A. What does "two for two" mean? Is it a "two tries for two wins"? (is that a right way to rephrase it?)
In this context, "two for two" can be rephrased as "two wins in two attempts" (or "two wins out of two attempts").
I would not say, "Two tries for two wins." That sounds like something you'd say before an event, when you are going to get two tries.