Drawing two cards from a deck of 16 (4 ranks and 4 suits)

For the first question, the easiest way to calculate the probability of not pulling a diamond card, then subtracting that from 1.

The probability of not pulling a diamond card is $$\frac{12}{16}*\frac{11}{15}=\frac{132}{240}=\frac{11}{20}$$ $\frac{12}{16}$ because that's the amount of cards that don't have diamonds, and $\frac{11}{15}$ because after pulling a non-diamond card, you will have $\frac{12-1}{16-1}=\frac{11}{15}$. Then, $$1-\frac{11}{20}=\frac{9}{20}$$

The second and fourth questions you have down, and for the third question just refer back to how I solved the first question.

For the last question:

What is the probability you don't get a Diamond card and you get two cards of the same rank?

You can do this most easily directly. The first card must not be a diamond, but can be anything else: 12/16 or, simplified, 3/4.

There are only two possibilities for the second card: the two non-diamond cards of same rank as the first card, out of the fifteen remaining cards. 2/15.

3/4 * 2/15 = 1/10