Technically, the Chicken came before the Egg. First of all, "C" is alphabetically superior to "E", therefore Chicken comes before Egg in all proper dictionaries.
Also, according to Genesis 1:24-25
"Then God said, "Let the earth bring forth all kinds of living creatures: cattle, creeping things, and wild animals of all kinds." And so it happened:
God made all kinds of wild animals, all kinds of cattle, and all kinds of creeping things of the earth. God saw how good it was."
Note it says animals, not eggs or seed of animals. Therefore, the Chicken was created before the Egg. That being said, it is proper to address the question as, "Which came first, the Chicken or the Egg?" as opposed to "Which came first, the Egg or the Chicken?". Thereby, since the proper manner of asking the question already gives due honor to the Chicken's superiority to the Egg, you have already answered the question through the question itself.
As to your other question, install the processor and the cooler first. Especially if it's the default/stock Intel heat sink. You need to make sure those push pins pop through all the way and are snuggly in place before you install the motherboard. This will save you some hassle later.