It sounds like your daughter most likely simply has a cold virus. Colds are very common in babies this age, often occurring multiple times per year. Having multiple colds does not indicate that your baby is "weak" or otherwise has some underlying condition.
In general, cold symptoms will gradually improve over 7-10 days. As long as your daughter is gradually improving, you should not worry
too much. On the other hand be on the look out for concerning symptoms. For example if your daughter is having high fevers, if she is breathing rapidly or if her chest looks like it is sucking in, if she is unable to drink liquids or has stopped urinating, or if she looks lethargic these are signs that you need to be reevaluated by your primary pediatrician
or go to the emergency room for an evaluation.
If none of these things are going on, then it is unlikely that you need to have an X-ray
or otherwise be worried. It is also unlikely that you need to give the baby this medicine, which is a cough suppressant. In general, most pediatricians recommend against using cough suppressants and similar medicines in young babies like your daughter.