Eating ajwa dates in the morning

Q:

1. It is reported in the Hadith that, “He who has a morning meal of seven ajwa dates will not suffer harm that day through poison or magic.” (Bukhari, Hadith 5327 and Muslim, Hadith 3814)

2. Hazrat ‘Aisha (Radiyallahu Anha) reported Allah’s Messenger (Sallallahu Alaihi Wasallam) as saying, “The ajwa dates of al-‘Aliya contain healing, and they are an antidote (when taken) first thing in the morning.” (Muslim, Hadith 3815)

I have a question regarding the two Ahaadith above. Do ajwa dates have to be taken at fajr time which is morning time Islamically, because these Ahaadith clearly say that it should be taken in the morning, or is any time during the day okay, provided it's the first thing eaten? If it has to be taken first thing at fajr time, does your stomach need to be empty? The reason for me asking is that I usually eat a heavy dinner at late night and my stomach is still full at fajr time. Also, what are Aliya ajwa dates and do you know where Aliya Ajwa dates are sold? 

Ajwa Dates

Q: Can you please explain what exactly Ajwa dates are, and give a brief history on it? Also, are all the dates in Madina referred to as Ajwa Dates?

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