What kind of pillows are there that are similar to down pillows?
There are many synthetic choices in the market today. Pillow similiar to down ones are probably just labeled "synthetic down pillows". Your best bet would be to go to Target or Walmart.
How can I pack my big comfy bed pillows for a flight?
If your carry on bag is big enough, you can use one of those roll-up bags that squeeze the air out of the pillow. I carry a pillow (although a bit smaller than my bed pillow) on all my flights. Here's the bag:
Where can I purchase dakki body pillows?
I found a body pillow at Bed, Bath & Beyond. I found a pillow case for it at Walmart!