The category of compact metric spaces

A rigorous proof that the infinite product of countably many 2-point spaces $A_n = \{0,1\}$ does not exist: (Really, just an elaboration of Martin's comment.)

Assume that $P$ is such a product, then $P$ must be $\prod_n A_n$ as a set. Let $e_n$ be the distance (in $P$) between the $0$-function and the function $x_n$ that takes the value $1$ only in the $n$-th place. The sequence $(e_n)$ must converge to $0$, because the $x_n$ will converge to the $0$ function. (They must converge by compactness, and the continuity of the projections means that all entries of the limit must be $0$.) [EDIT: As Martin Brandenburg pointed out, compactness is not enough to show convergence of the whole sequence, but only of some subsequence $(e_{n_i})$. For notational simplicity, assome $n_i=i$ for all $i$; otherwise restrict attention from $(x_n)$, $(e_n)$, $(d_n)$ to the respective subsequences $(x_{n_i}$, $(e_{n_i})$, $(d_{n_i})$ from now on.]

Let $(d_n)$ be a sequence of real numbers converging to $0$, but more slowly than $(e_n)$, i.e., $e_n = o(d_n)$. (E.g., $e_n=\sqrt{d_n}$.) Let $Q = \prod A_n$, but with the following metric: $d(x,y) = e_n$, if $x$ and $y$ agree on the first $n$ values, but not on the next one. Project $Q$ to each $A_n$ $-$ these maps are Lipschitz-continuous. So they factor through a map from $Q$ to $P$. Now this map must be the identity (apply the forgetful functor); but points with distance $e_n$ are then mapped to points with distance $d_n$, which is not possible for a Lipschitz function.

Infinite coproducts do not exist. For instance, suppose $C$ were a coproduct of countably infinitely many $1$-point spaces. The inclusion maps of these $1$-point spaces then give a sequence $(x_n)$ of points of $C$. Since $C$ is compact, some subsequence $(x_{n_k})$ must converge. But now you can just take any compact metric space $Y$ with a sequence $(y_k)$ which does not converge. By the universal property of $C$, there must be a Lipschitz map $C\to Y$ which sends $x_{n_k}$ to $y_k$ (and the $x_n$ which are not of the form $x_{n_k}$ to wherever you want in $Y$). But there does not even exist such a map which is continuous, since $(x_{n_k})$ converges and $(y_k)$ does not. (This example also shows the forgetful functor to Sets has no left adjoint, since such a left adjoint would need to send $\mathbb{N}$ to such a coproduct.)

Coequalizers do exist. Indeed, it is easier to just show that quotients by arbitrary equivalence relations exist. Quotients by arbitrary equivalence relations exist in the category of metric spaces and Lipschitz maps with Lipschitz constant $1$ (for instance, see this answer of mine on MO), and a quotient of a compact metric space is compact since the quotient map is surjective. These quotients have the same universal property when you allow Lipschitz maps with arbitrary Lipschitz constant, since a map $f:X\to Y$ is Lipschitz with constant $K$ iff it is Lipschitz with constant $1$ when you rescale the metric on $Y$ by $1/K$.