I would say rethink using flash. If the reason it's discouraged is green-eye and that deer-in-the-headlights look, it's simply a matter of learning to use flash correctly, whether that's bouncing an on-camera flash, or taking the flash off-camera.
Just my opinion, but I think your concerns about the power/light output from low cost continuous lights is a good one. Generally, lighting effects are achieved by creating different levels of lighting and the more light output a light gives you, the greater the differences you can make and the more flexibility you have in the image you want to create. Continuous lights that allow you to do large levels of difference are liable to either be expensive or very hot, and it's if you're primarily doing video that you really need to consider them.
Flashes actually are the low-cost way to get a lot of light. It is more difficult to visualize than a continuous light, which will be WYISYG :), but it's also a skill that can be learned. The most commonly recommended website that can walk you through learning about studio lighting is the Strobist. The Strobist began as a blog in 2006 by the then-photojournalist, David Hobby. He had had great success using hotshoe flashes in off-camera studio lighting setups that were small, low-cost, and portable, to shoot editorial assignments, and he began the blog to teach other photojournalists how to do this. And he ended up teaching, well, basically, a lot of digital photographers on the internet, too. They use a Flickr group to share ideas and photos, and you can see the dog shots from the Strobist Flickr group to get an idea of what and how it can be done (the group requires that you include lighting information in your descriptive text of the photo).
Umbrellas and softboxes are merely modifiers you add to a light. They should not be an integrated part of the light. They have different characteristics, but both soften the light. An umbrella is typically cheaper, but makes it more difficult to control the spill of the light (i.e., where the light doesn't go). A softbox gives you more control over the spill and can help you shape an edge/falloff to the light. Which one you want depends on the image and mood you're trying to create.
If you're using a hotshoe flash, an umbrella swivel attaches the flash to the stand, as well as giving an attachment point for an umbrella. A studio strobe will connect directly on the stand, and may provide the umbrella attachment point. Studio strobes are bigger and more powerful (i.e., even more flexibility than a hotshoe flash), but typically need to be plugged into an outlet, rather than using batteries. They're bigger and heavier, and can be more expensive than a speedlight (hotshoe flash).
I will also add that the Strobist way of shooting with hotshoe flashes became highly popular because the gear is inexpensive. You don't need to use OEM (i.e., Canon/Nikon) flashes. There are some very cheap low-cost manual-only flashes that can be used this way, and as long as you only need them for occasional hobbyist usage, rather than hard constant professional usage, they might suffice. For example, a Yongnuo YN-560III (which has a radio receiver built in) can be purchased on B&H at the time of this writing for US$66.