The reason why a doctor told you to take half the pill if for a smoother start. You will eventually increase dosage but it is more easy on your neurotransmitters to start slowly and therefore, it is my opinion, you should take half the pill for a week. Generally i would be telling you listen to your doctor but there seems to have to much contradictions so I'm unable to do so and will use my knowledge based opinion/suggestion on the topic.
When starting a new drug, especially with antidepressant, which cause significant changes in your neurotransmitters, Lexapro is affecting the serotonin neurotransmitter, your body needs to adapt to these changes and this is why you are experiencing symptoms you mentioned and it is also why the drug should be started slowly then gradually increased.
You mentioned dilated pupils, very common symptoms when starting a drug. One thing that is less common is anisocoria which means one eye will have a different pupil size than the other eye. If you experience anisocoria you might want to check with your neuro to rule out more severe health conditions.
Your body can take up to 1 month to adapt to the new drug and leave you symptoms free but everyone is different and one drug doesn't fit all so if the symptoms persist you might want to check this issue with you doc and he might have you to try a different drug.