Hey, you are fine. Sometimes its a lot worst when you over think it. BUT you are in an academic setting, this is where you learn and explore. Your PI hopefully is understanding and just wanted to see what you are capable of. Sure you didn't get it how you wanted, maybe complete botch but be honest and let your PI know. You only grow from there!
Ask letter writers a few months before the apps are due. If you tell them too far in advance, they'll forget. I suggest asking after apps open, in case they want to do it right away. Make sure to follow up a couple weeks before apps are due. Don't expect them to do them in a timely manner.
Shouldn't matter if you spell them out. If you have the space, go for it, otherwise, they know their own program's name.
I recommend making it a CV. Consider what's most important to them and order your sections that way. Mine was: education, research experience, publications and abstracts, other experience.
Bak3rm3 got a reaction from Sigaba in Day 1 of PhD and already embarassing myself in lab