Can an FO formula with infinite length be expressed with an infinite set of finite formulas?
You are getting at infinitary (first-order) logic. While a well-defined (and natural, and extensively studied) concept, it is quite different from ordinary first-order logic. Before diving into more detail, let me point out the main issue with your suggestion:


  There are "infinitely long" expressions which aren't just infinite conjunctions of finite expressions.


That is to say, your idea


  Since any formula has an equivalent formula in CNF, if we transform a formula of infinite length to some CNF, and then break the conjunct up into clauses and put them in a set, is it then expressed with an infinite set of finite formulas?


is incorrect. One way to see this is with the compactness theorem. For example, suppose I had a set $\Gamma$ of (finitary) first-order sentences which expressed connectedness. Then consider the expansion of $\Gamma$ to include sentences saying "$c$ is at distance at least $n$ from $d$," for each natural number $n$, where $c$ and $d$ are new constant symbols. By the usual compactness argument, this has a model, but (the reduct of) this model isn't a connected graph; contradiction.

The usual manipulations we perform with finitary formulas don't behave intuitively in general when we look at "infinitary" formulas, and in fact infinitary logic (see below) is a very different beast from its finite counterpart.



Having said what infinitary logic isn't, let me end by saying a couple sentences about what infinitary logic is.

For a start, let's look at the simplest nontrivial example: $\mathcal{L}_{\omega_1,\omega}$. This is smallest class containing all first-order formulas, closed under (a) quantifiers and (b) countably infinite conjunctions and disjunctions. This logic is much stronger than first-order logic - for example, connectedness (in the language of graphs) is definable by an $\mathcal{L}_{\omega_1,\omega}$-formula, namely

$$\forall x,y[\bigvee_{n\in\mathbb{N}}\exists z_1,...,z_n(z_1=x,z_n=y, z_1Ez_2E...Ez_n)].$$ This is a relatively simple infinitary formula, and there are much more complicated ones we can consider; there is a rich (and non-collapsing) hierarchy of infinitary formulas, and in fact many fragments of $\mathcal{L}_{\omega_1,\omega}$ of mathematical interest (see e.g. this paper for a good starting point).

We can go beyond $\mathcal{L}_{\omega_1,\omega}$ by either allowing longer Boolean combinations, or infinitely long blocks of quantifiers, but as we go further up the hierarchy of infinitary logics things get progressively much nastier. $\mathcal{L}_{\omega_1,\omega}$ is quite abstract, but still "concrete" enough to be relevant outside of infinitary logic - see e.g. the notion of a Scott sentence, which is useful in computability theory, classical model theory, and set theory.

There are several texts on infinitary logic; I recommend Marker's recent book as a good starting point, and Barwise's old book for a more in-depth treatment of the "local" flavor.