Polar bears eat ice seals (and some others). Could they learn to eat Antarctic seals instead? Maybe. Would they eat penguins? Maybe. In both cases, the hunting for them would be different, as the habitat isn't the same and the prey animals have different habits.
This would change the balance of seals (or penguins) and affect their prey. And their other predators would have less food.
Can you bring down the Arctic ice seals and other seals? Sure. But then they would compete for food with the existing seals (assuming the food was close enough to Arctic food for them to make the transition).
This changes the fish and mollusks and sea vegetables and plankton and pretty much everything. How does the forced migration change the gut biome of the polar bears? And of the other Arctic animals. Will their droppings change the formation or growth of the microscopic organisms and other small lifeforms that various Antarctic animals depend on?
So the answer to your first question is: none. I'm trying to think of any transplanted species that thrived but did not cause vast changes to the existing ecosystem. Even bees and earthworms have done this in the Americas. Not all change is bad for the ecosystem as a whole, but it will harm (or even wipe out) some species.
As to your second question... Polar bears are solitary and don't live in groups, so the numbers you need would be for reproduction. Anything up to 20-25K (which is the estimated total wild polar bear population). There are 15 groups of polar bear populations, which leads me to believe that 1000 polar bears would be a reasonable relocation group size. Most polar bear litters are 2 cubs and a female has about 1 litter every 3 years. The smallest home range area for a group appears to be 50K square kilometers.
I don't know the answer to your third question, but I would guess that a good time would be after they've done some heavy feeding in preparation for winter.
Sources: There is lots of similar info in many places over the internet. Some good basic facts are here: https://seaworld.org/animal-info/animal-infobooks/polar-bears/habitat-and-distribution