Re: Sleepy baby, not gaining weight.
Sk, I hope your baby is doing much better. My baby was born with Down Syndrome and was also preemie, and he had extreme sleepiness. I was told that this could be normal for preemies and also for babies with Down Syndrome. Your baby is older now so maybe she's outgrown this. Also, sometimes if there is a heart defect this can also affect their alertness or energy level.
The sucking and swallowing problems can continue for several months, so like DivineEntity mentioned, you want to make it as easy as possible for her to get calories. Sometimes breast feeding can be difficult for a Down's baby because of the hypertonia. Hypertonia is low muscle tone and the baby can have it in any place in their body, including around their mouth. My baby could not breastfeed although I tried for a long time; he just didn't have the energy.
Physical and occupational therapy is so critical at this stage and they can help with the hypertonia.
Also you may want to make sure the hospital or pediatrician has done a comprehensive workup of blood tests on your baby just to make sure there are no other problems that Down Syndrome children can have, such as celiac disease or thyroid problems. And don't be afraid to ask questions! Ask lots of questions, and if your doctor doesn't know the answers and doesn't research them or seems disinterested, you may need to look for another doctor or ask to be referred to a specialist. Educate yourself so you can help your child.
I know it sounds intimidating at first, there's so much to learn. But just keep at it, things will get better! One day your baby will thank you.