I use stories like these to develop intuition... or perhaps to destroy it. I have my own answers in mind, but I want to see if I have made any mistakes...
You are standing at the origin of an "infinite forest" holding an "infinite bb-gun." The "trees" in this forest are at the lattice points all around you. (The lattice points are like those on graph paper and they align with the cardinal directions: N, S, E, W.) The "forest" is Euclidean in the sense that the trees have no width. To hit a tree with your bb-gun you must aim perfectly at it.
You would, for example, hit a tree if you fired the gun due north, south east or west. (Your bullets also have no width.)
A. You fire the gun in an arbitrary direction without bothering to aim. What happens?
B. You get a new bb-gun and the bullets have a little width to them. ($\delta$?) You fire the gun in an arbitrary direction without bothering to aim. What happens?
C. All of the trees are removed that have coordinates whose absolute values are not perfect squares. (So, only points such as $(25, 100)$ and $(4,-1600)$ remain.) Again you use width-less bullets. You fire the gun in an arbitrary direction without bothering to aim. What happens?
D. Again, only with perfect squares, but now the bullets have width. What happens?
A., C. The probability that you hit a tree is $0$. This is equivalent to the statement that the set of angles at which you can see a tree is of measure zero in $[0, 2\pi)$ with the Lebesgue measure, which follows because it is countable.
B. The probability that you hit a tree is $1$. By Dirichlet's approximation theorem, for any real $\alpha$ there exist infinitely many pairs $p_n, q_n$ of integers such that $|\alpha - \frac{p_n}{q_n}| < \frac{1}{q_n^2}$. It follows that the distance between the points $(q_n, q_n \alpha)$ and $(q_n, p_n)$ is at most $\frac{1}{q_n}$, and since the sequence $q_n$ tends to $\infty$ it follows by letting $\alpha = \tan \theta$ where $\theta$ is the angle at which you fired that for any bullet size $\delta > 0$ we will hit a tree at $(q_n, p_n)$ where $q_n > \frac{1}{\delta}$.
D. The probability that you hit a tree is still $1$. This is a consequence of the following theorem, which I just learned about by asking this MO question.
Theorem (Khinchin): Let $\phi(q) : \mathbb{N} \to \mathbb{R}$ be a monotonically decreasing function. For almost all real numbers $\alpha$, the number of pairs of positive integers $(q, p)$ satisfying
$$\left| p - q\alpha \right| < \phi(q)$$
is infinite if $\sum \phi(q)$ diverges, and finite if $\sum \phi(q)$ converges.
In particular, taking $\phi(q) = \frac{1}{q \ln q}$ (the sum of which diverges, for example by the integral test), we get that for almost all real $\alpha$ there are infinitely many solutions $(q, p)$ to
$$\left| \frac{p}{q} - \alpha \right| < \frac{1}{q^2 \ln q}.$$
Now let $\alpha = \sqrt{\tan \theta}$. With probability $1$ there will be infinitely many $(q, p)$ satisfying the above condition. Then
$$\left|p^2 - q^2 \tan \theta \right| < \frac{\left| \frac{p}{q} + \sqrt{\tan \theta} \right|}{\ln q}.$$
By taking $q$ large enough so that the RHS is less than $\frac{\delta}{2}$ it follows that a bullet shot at angle $\theta$ will hit the tree at $(q^2, p^2)$. (I'm only working in the positive quadrant but the generalization to the other quadrants should be clear.)