Problem 12: The problem says that the sun is setting at a rate of 1/4 deg/min. In reference to the picture given in the book, does that mean that the angle where the dotted line and the ground meet (which I'll call theta) is decreasing at a rate of 1/4 deg/min?

Yes, that's correct, but there is one glitch. You need to convert to radians/minute (otherwise, our derivatives will be a mess). So, multiply by $\pi/180^{\circ}$. Also, because the angle is decreasing, the corresponding derivative is negative.

If so, the formula we use would be tan(theta) = opp/adj, right?

Yes!

In this case opp would be 25 and adj would be 50.

Yes…but be careful! The 25 is fixed for all time, but the 50 is for one specific moment in time. So, label the length of the shadow with a variable, say $x$.

I did the problem this way and got an answer that's far from correct, so maybe I'm setting it up wrong.

Perhaps your only error was the degrees to radians issue.

Problem 14: It seems like more information is needed here but I really don't even know where to start. Staring at my picture doesn't help.

You should have two right triangles, one inside the other. The height of the big triangle is given by the light post and the height of the other triangle is given by the height of the woman. The equation you need to write down involves similar triangles. One additional hint is to label the distance from the pole to the woman with a variable and the distance from the woman to the tip of her shadow with a different variable, so that the total length of the base of the large triangle is the sum of the two variables.