When you say "upgrading" do you mean conversion or do you mean new application?
Here is the difference:
Conversion -- You call Amex and have them covert your current account to a Blue Everyday. This (in my experience) does not require a "hard pull" on your report. This route however usually also does not qualify for introductory rates like 0% for 6 months.
New application -- You apply for a new card via normal means. This usually results in a "hard pull", but will get you into rates and promotions.
I found this link in an older post of mine. Depending on you spending you can get better rewards from your current credit card than the Blue Cash Everyday. Here is a link to an older Amex rewards calculator for your card:Amex Blue Cash Rewards Calculator
Should be easy enough to compare the old and the new as the new does not have a tier structure.
The second question is a bit harder to answer. It all depends on your spending. Yes there are cards that will give you more cash back if you spend on categories they promote. Question is do you spend enough in each category to make it better for you?