All of the area's that you mentioned above are good area's. From personal experience Capitol Hieghts and Eastern Market are both good areas. But one quick turn off onto another street and you could land your self in a not so good neighborhood

! All of the other area's are very safe in my opinion.
Private vs. Management
I have actually done both, I prefer management. My reason is, is that if something breaks you call the maintance they come and fix it. You dont have to wait forever for them to come out and fix it. Management, if they need to come into your apartment for anything they will first send around a letter telling you that they will be entering the apartment. You dont just wake up and find your landlord in your apartment fixing something. For me with management there is more of a level of courtesy than with private. I dont want to scare you away from the thought of private because it could have just been my landlord ya know!
Good luck with your search and I hope that my info could be of some help to you.