My syringe came with some baby medicine, I bought the medicine purely for the syringe and the rest went to...someone with babies!!!
You do indeed stuff the woolies into the syringe, on mine, there is just a hole at the bottom of it, I tried to get used syringed woolies out and realised I had no idea how!! So I got some tweezers and stuffed erm into the hole which widened it and kinda damaged the syringe but it still works!! Hmm maybe I will use a paperclip or something with my next syringe...