Will the new promotion boards look at my bad paperwork, reflecting stuff I did as an E-5, to determine my eligibility for promotion to E-7?

Yes, those are labeled as a "-R" in your board file and they are required to look at it. After a year you can request to have it moved to your restricted file. But the board members as re required to read them.

In the old days..I used to just remove my bad paperwork.. normally during an SRP. Once in the Marines I took out the charge sheet.... from my Art 15 proceeding.. The LTC yelled at the Legal Officer who yelled at the Legal Assistant. They searched in vain.... case dismissed.. but they made me paint the admin offices after hours and I found the hidden extra key to the keybox... my anger issues kept me an E-4....

There are a number of good answers here but I will add mine as well, Since your record is a reflection of you and your service it is the tool used in making promotion selections, everything in it is open for review to some degree, However what you have done after the bad paper can and will off set to some degree that problem. It has been pointed out it does contain a Restricted area for certain items, In your case something like the Bad 1059 is usually able to be moved here when you show the deficiency has been overcame, I use this as an example because I had a bad one for one school as an E5 also, I attended as an E6 about 16 months later and the 1059 was a total opposite picture of my time there, I then also finished my Advanced Course a short while later as well as several others of a speciality nature, When My record went before the E7 Board the first time I was not Selected, however a year later and after an Involuntary Reclassification due my MOS being phased out I was Selected with a sequence number of 7 out of 28 in the new MOS. You need to dig in and learn how to make your record the best possible reflection of you using the guidelines long ago established to not allow good people to be sidelined by a relatively minor mistake while they were still in the growing phase of Military Service.