That's because like in abstract algorithmic descriptions, there is no distinction between code and data, in the physical world, there is no distinction between interpreted code and interpreter, it's just a chemical reaction between the two.

(That's true for computers (on semiconductor basis) as well, is the behaviour of an addition in the ALU or in the bit pattern to select the instruction? It's both, the ALU alone would exhibit all kind of operations (including unintended/not-meaningful ones), the bit pattern alone could trigger any kind of operation depending on the circuit it occurs in. But if both happen to occur at the same point in spacetime, the physical laws apply and the bit pattern emerges, that humans would call the addition.)

The DNA doesn't fold itself. There is an entire collection of proteins and enzymes that maintain it (example: histones and proteins that adjust histones). Some DNA sequences do affect that folding, but not in an "active" way.

There are a very small number of DNA enzymes produced in labs, but there are no endogenous DNA enzymes (that anybody has found so far). Instead, it seems like life is much more based on RNA enzymes and proteins (the core of the ribosome is an RNA machine, decorated by proteins that enhance its speed and reliability).