Why do videographers have rectangular lens hoods available?

Primarily because videographers frequently mount rectangular filters such as NDs and ND grads, and for the ability to use flags or barn doors to more finely control light blocking.

Light control is more important in videography than in photography, because one exposure control variable, shutter speed, is not a free variable under the videographer's control (they are limited by the frame rate for the video). The adjustable barn doors / flags correspond to the image's aspect ratio crop. Of course, straight-edged barn doors that correspond to the rectangular aspect ratio allow for tight light control, as well as being easy and inexpensive to manufacture. Rounded or curved barn doors with adjustable cone shapes are a lot more complicated and expensive to make, and are not useful because they don't match the aspect ratio crop.

Adjustable barn doors are not as useful in photography. The photographer's position is typically more fluid / transient, and even when it's not, shading and flagging needs are usually only for a few shots, and can be easily controlled by an intern with gobos or block cards, or stands with clamps to hold the gobos.

Do they only use Internally-focusing lenses?

Pretty much, yes. But not necessarily exclusively. All of the cine lenses I've seen are internally-focusing (but my experience with cine lenses is rather limited). Non-IF lenses are undesirable in videography because everything in front of the lens (matte box, filters, etc.) have to move in conjunction with focus changes. And because of the weight of matte boxes and filters, they aren't mounted onto the lens; they are mounted on the video accessory rails in front of the lens.

Lastly, why don't DSLRs use rectangular hoods?

Most photographic lens hoods are round (or tulip shaped) for compactness. A round lens nestled in a round hood in a round or round-ish lens pocket or camera bag compartment allows us to be more space-efficient with our gear.

Note that some photographic lens hoods are rectangular. While not DSLRs, many hoods for rangefinder camera lenses are rectangular. As WayneF notes in his answer, the tulip-shaped hoods on many DSLR lenses are the approximate intersection line between the projected rectangular field of view and a cylinder or cone.

In the context of slide-in square or rectangular filters, rectangular hoods are not very convenient in photography. Videographers never have to worry about orientation: their video is always presented in landscape orientation1. Photographers frequently change between landscape and portrait orientation. For shooters who use them, directionally-oriented filters (such as polarizers and ND grads) need to be large enough to be usable for either orientation. Note that ND grads and polarizers must maintain the same position/orientation with respect to the scene, regardless of camera orientation. As I mentioned, that isn't a concern in videography (for the most part). But in photographic uses, that requirement of orientation-independence strongly implies rotational symmetry, i.e., circles. That doesn't require filters to be circular; rather, it implies that if a filter covers a certain circular area, it is guaranteed to cover the lens's entire image circle, regardless of camera orientation. This is one of the reasons you don't typically see hoods mounted in front of square/rectangular filter holders on photographic cameras. But you often see matte box filter holders with barn door, bellows, or other rectangular hoods in front of them.

For photographers who do need or want rectangular hoods or matte boxes, they are available. Those users are just a relatively small portion of the DLSR-wielding population.

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1 : I'm not talking about people who take vertical videos with their smartphones.