Here is my take on Elia’s medical issues. They could all easily be traced to her prematurity which can cause a variety of long-term problems. First of all, a preterm baby in Westeros would have to be over 32 weeks because a younger baby would not have lungs developed enough to survive without modern medicine. In fact, a baby under 34 weeks would probably not have enough lung power to make it. This means Elia, if she were considered very premature, was probably a 34 to 36 weeker. So she missed out on a third to a half of the third trimester.
EVERY preemie, even those born close to term has lungs that are not as mature and resilient as a full term infant’s. So if she gets any respiratory infections in the first year or so of her life, she’s gonna get a lot sicker than a kid who cooked for 40 weeks would. This can lead to long term scarring in the airways and lifelong respiratory diseases like asthma. There are a few herbals that legitimately help with bronchispasm which theoretically could have been known to maesters and healers which would have helped Elia ease her symptoms if she had asthma and allow her to live into adulthood, etc., but nothing like the bronchodilators or steroids we have now–so she definitely would have had restricted activity level and pregnancy would have been difficult.
One important thing that takes place late in pregnancy is the transfer of minerals like iron and calcium from mother to baby. So all preemies tend to be born with some degree of deficiency in these things. Iron deficiency causes anemia and calcium deficiency causes problems with weak bones. All preemies receive supplements now. But of course supplements wouldn’t exist in Westeros so it would be hard for a preterm baby to climb out of the hole they start and where these minerals are concerned.
Her small size would put her in a nutritional hole as well. Smaller bodies have a greater surface area compared to their total mass so simply maintaining body temperature requires a lot of energy. That’s why premature babies get put in Isolettes, once called incubators. It helps decrease the caloric requirements of the babies which is important because they aren’t great at eating. Being born in Dorne was probably a help to Elia’s survival at least because at least it’s easier there to keep warm there than in Winterfell! Except for Catelyn’s chambers maybe. :-D
If Elia couldn’t latch to a breast, she had no infant formula, feeding tubes, or even bottles with rubber nipples. The best that could be done is for someone to express breast milk and use something like a rag to drop it into her little mouth. You could possibly maintain her with that, but you are not catching up on calories, minerals or other nutrients.
So she’s set up for long term poor growth, anemia, brittle bones and a whole slew of health problems related to early childhood poor nutrition.