Why do animals with eyes nearly always have exactly two, and did predators get them first?

The answer to the number question is much less about optics than you would expect: it is inherited body plan, not optimisation.

Most animals you are thinking of are bilaterally symmetrical — a left side and a right side, an ancestral arrangement shared by an enormous group of animals. Paired structures are the default in that body plan: two eyes, two nostrils, two ears, limbs in pairs. Two eyes is not a solution somebody converged on; it is what you get when the ancestor had a pair and every descendant inherited the arrangement.

The evidence for this is that animals outside that body plan do it differently. Spiders have eight, many insects have compound eyes plus simple ones, scallops have dozens along the mantle edge, box jellyfish have clusters of them. Where the constraint is absent, the number varies enormously.

So the question "why not four" has the slightly deflating answer: because our ancestors had two, and there has been no pressure strong enough to justify rebuilding the head.

The other half of the answer is that eyes are expensive, which is why nobody adds spares.

Measurements in fish suggest that vision can consume a substantial fraction of resting metabolism — the figures are striking, and the cost is highest in small animals. Neural tissue to process the image is costly, the eye itself is costly to build and maintain, and it is a vulnerable opening in the body.

The evidence that this cost is real and continuously felt: cave animals lose their eyes, repeatedly and independently, in lineage after lineage. If eyes were cheap, they would simply be retained and unused. They are not, so they are dismantled within surprisingly few generations.

Given that, adding a third eye would need to pay for itself against an already expensive system. Two is what the body plan supplied and evolution has generally spent its budget improving those rather than adding more.

On predators first: the reasoning is sound and the evidence is suggestive rather than settled.

The hypothesis that visual predation drove a major burst of animal diversification is well known and much discussed — the argument being that once something could see prey at a distance, everything else had to respond, and hard parts, mobility and eyes proliferated together.

What complicates the neat story is that simple light detection is far older and much more widespread than image-forming eyes, and it serves purposes with nothing to do with hunting: day length, depth, orientation to light, avoiding damaging radiation. So the raw material was lying around for a very long time before anyone hunted with it.

The likely picture is that predators drove the transition from detecting light to forming images, rather than inventing photoreception. That is a more modest claim than the popular version and it fits the fossil and genetic evidence better.

One thing worth adding on depth perception, since it is the answer you started with: it is not the main reason most animals have the eye placement they do.

Compare a predator with forward-facing eyes and a large overlapping field, to a prey animal with eyes on the sides of the head and nearly panoramic coverage but little overlap. Same number of eyes, opposite priorities — depth in front versus detecting movement anywhere.

So the pair is inherited, and what varies is where they point. That variation is where the actual optimisation shows up.