Brief Answer:
may need  surgery after sometime
Detailed Answer:
Hi XXXXXXX 
I know u are very concerned but as a doctor I am advising you the following,
Do not worry.
This picture is very common in 3 year olds.!
The 
adenoids do not go and come ,
They are there always,get enlarged with infections and when enlarged Press on the upper airway .
Any child from 2-5 years of age has some adenoid and tonsil enlargement ( called lymphoid enlargement) which will regress with age.
Mouth breathing may be from adenoids or blocked nose or by  habit !!!
I understand you do not want surgery  and I agree with you as adenoids are not that big in the X-ray also.you  can wait for a year or two and watch.
See that there is no  nasal block - if there is you can use pediatric nose  drops like 
otrivin pediatric when necessary.
Antibiotics are not needed Evert time she gets 
cold and  cough!!!!
Antibiotics are needed only if there is 
fever,yellowish phlegm,
follicular tonsillitis,etc.
Levocetrizine is antihistaminic and can be used .
Montelkeukast is anti allergic and can be given a trial.
Why 
steroid spray when lungs are clear?
Is it  for nasal allergy? ( not for adenoids surely) steroid spray can be used for a limited period after all other measures have failed)
Try to accept minor symptoms as if you pressurise doctors will go for stronger medicines.
Viral illnesses do take time to clear.
I am sure your child will improve in a few months with the change in weather.
Try to prevent all allergens(. Dust in the house, bedroom,crowded places. known flood allergy ( you will find by trial and  error)
Proving infection indirectly is by checking blood counts .
Good luck
God bless
Dr Uma