Does the infinite graph of diagonals have an infinite component?
The probability is 0.

This follows from the following theorem (see Theorem 5.33 in Grimmett's Probability on Graphs, http://www.statslab.cam.ac.uk/~grg/books/USpgs-rev3.pdf):

Theorem Consider bond percolation on $\mathbb Z^2$, where each edge between lattice points is open with probability $\frac12$. The probability that the origin is in an infinite connected component is 0.

We can reduce from our problem to this problem: basically it is just 2 disjoint (but dependent) copies of bond percolation on $\mathbb Z^2$. Consider the configuration $D_1$ where the edges form an infinite lattice of diamonds containing the origin. If we flip all edges, we get another infinite lattice of diamonds $D_2$. Consider the intersection of the actual configuration with $D_1$ and with $D_2$. Each of these is exactly the model of bond percolation on $\mathbb Z^2$, just rotated $45^{\circ}$. The probability that any point is in the infinite cluster is hence 0 (No edge in $D_1$ can be connected to an edge in $D_2$.). 

To conclude, note that the sum of a countable number of events with probability 0 has probability 0; sum over the probability that any lattice point is in an infinite cluster.

(The existence of arbitrarily large components is a red herring. One should fix a point, and ask if it's in an unbounded component.)