For the first question:
Let's think about just selecting the first digit. There's a probability of $\frac 1 7$ of selecting $7$, $\frac 1 7$ of selecting $6$, and $\frac 1 7$ of selecting $5$. In any of these cases, the resulting number is greater than $4200$. Add up these probabilities: $\frac 1 7 + \frac 1 7 + \frac 1 7 = \frac 3 7$.
Then there's also a $\frac 1 7$ probability of selecting a $4$. Then we need to pick the second digit. If we pick a $2$, then we note that our number will still be greater than $4200$, because picking zeroes for the last two digits is not an option. So the probability of picking one of $\{2,3,5,6,7\}$ is $\frac 5 6$, and any of these will yield a number greater than $4200$. Note that we can't pick $4$ as the second digit because we're selecting from $D$ without replacement.
What's the probability of picking a $4$ and then picking one of $\{2,3,5,6,7\}$? They are independent, so by the multiplication rule we can just multiply them: $\frac 1 7 \cdot \frac 5 6 = \frac 5 {42}$.
So the total probability is $\frac 3 7 + \frac 5 {42} = \frac {18} {42} + \frac {5}{42} = \frac {23}{42}$.
For the second question:
Probability of seeing at least two differently numbered faces $= 1-$
Probability of all four tosses showing the same face.
How do we calculate the probability of all four tosses showing the same face?
Well, suppose we said that all four tosses had to show $6$. The probability of that for a single toss is $\frac 1 6$, and by the rule of multiplication for probabilities of independent events, the chance of tossing four $6$s is $\left(\frac 1 6\right)^4$.
And that's the exact same for all other faces $1,2,3,4,5$. There are six such faces in total, so the probability of all four tosses showing the same face is $6 \cdot \left(\frac 1 6 \right)^4 = 6 \cdot \frac 1 {6^4} = \frac 6 {6^4} = \frac 1 {6^3}$.
(If you want more intuition on this -- draw out the table of probabilities for two rolls of the dice, and note that the diagonal is exactly six events.)
So the probability of seeing at least two numbered faces is $1 - \frac 1 {6^3} = 1 - \frac 1 {216} = \frac {215}{216}$.
I think you meant $\ldots = 8.76$, which is indeed correct.
You're missing an $X_4$ in your sum:
$$E(X_1 + X_2 + X_3 + X_4 - H) = E(X_1) + E(X_2) + E(X_3) + E(X_4) - E(H),$$
although you did account for it in your calculation.
Best Answer
Hint: what is the probability that the same face pops up over the four rolls? The answer you seek is one minus this probability.
To attack this, consider what happens. You get a first roll and record what you get. The probability that the next 3 rolls give you that same face is...?