If Earth had a average below freezing level temperature would all lifeforms on Earth die? Except all the animals in already below freezing places like penguins and some bears.
Short answer: no.
Long answer: depends what you mean.
The hardiest creature known to man is the tardigrade, also called the waterbear. It can survive exposure to temperatures as low as −273°C (though it is my understanding that it must be warmer to actually function). And as you mention, there are plenty of other creatures that can survive at temperatures below 0°C. Humans can do it, too! …so long as they are prepared. Therefore, we know life can sustain at such low temperatures. So if your question meant to ask whether life could exist on an Earth-like planet with a low average temperature below freezing, then I think the answer is "yes".
However, perhaps you meant "Would everything on Earth die if Earth's average temperature were suddenly restricted to being below zero?" I think the answer is still "no": the animals that already live in such climates would continue to do live. However, there would likely be catastrophic interference with the global ecosystem. In the short-term, everything that lives near the equator would probably die. Probably even a significant portion of close-to-the-surface ocean life would die (though the deep-sea creatures might continue to survive, since the lower reaches of the ocean are heated from the earth's mantle and do not depend on the sun's energy at all for sustenance). This widespread die-off would impact the human population, and a significant portion of us would die (but, of course, some people might manage to live a more primitive lifestyle in the cold regions). So I don't think all life would die, but certainly the vast majority, and any surviving lifeforms would be very limited in their expansion capabilities.
(And honestly, it's possible that the global collapse of the ecosystem could have ramifications into the local ecosystems of the cold-dwelling creatures, too. If the impact were great enough, which it may be, then it could lead to the death of all macroscopic life. However, some microbes and bacteria might manage to find a way to live, somehow. Probably photosynthesis at first, and maybe evolve to live off of the waste we've generated over time. In any case, it would be a mass extinction event.)
The average temperature on Earth now is about 15°C. During a glaciation the temperature can be 5° lower, and ice sheets spread as far south as London.
Now the average temperature in london is about 10°, yet during a glaciation it get covered in ice, this is because there are feedback mechanisms. As the ice spreads towards the equator, it reflect light causing cooling. The polar regions tend to amplify climate change.
If the Earth were to cool by 15°, the Earth would still not be completely frozen. If animals had time to adapt there would continue to be complex life around the edge of the ice sheets. The question is would the temperature stop at 0. If ice sheets were very extensive, positive feedback could cause the Earth to continue cooling until ice covers the whole Earth (which requires a global cooling of more than 30°)
This is bad for life! Bacteria would survive, but it is not certain that anything else would. Polar bears need to eat seals, which need fish, which need krill, which would be killed by the frozen ocean: hence no polar bears. The cause of death would be the collapse of food chains.
This has (probably) happened before: about 700 million years ago, there were very extensive ice sheets. But back then there was little complex life.
So if the temperature drops to freezing it's bad, but survivable. If the temperature drops enough to freeze the whole earth, it's back to the precambrian.
If we look to the past we can only assume the all lifefoms on Earth would not die. Humans are capable of building shelters and using other resources to keep warm. Such as fire and, animal fur. Humans can also eat a great amount of varied foods. If bears and penguins are living then they also have a food source. Penguins eat fish, fish (other than carnivorous fish) would need a food source such as algae or seaweed. These sort of foods can be consumed by humans as well. Humans also could consume the bear, the penguin, and the fish. Modern man has a leg up on earlier man in many ways. For instance modern man can grow plants with artificial light. Modern man can create shellter from the cold but still allow the warmth of the suns rays to be utilized. Modern man can breed animals in sheltered areas. Modern man can utilize many different forms of life giving energy. So a resounding no is my answer.