Mature feathers have a completely hollow shaft, you can't make them bleed no matter how close to the skin you cut them. Growing feathers however still have a blood supply in the shaft and if you cut one of these blood feathers it will bleed. A lot. If you accidentally cut a blood feather, the best thing to do is to grab the feather at the base and pluck it out. Then apply pressure for a few minutes to stop/slow the bleeding. It might be dramatic, but a healthy bird will not bleed to death from a few damaged blood feathers
Personally, I prefer to clip both wings. Clipping one wing will throw her off balance, but she will still get some lift from the other wing and being off balance may make her more likely to injure herself in a crash landing. If a bird is still able to get lift with both wings trimmed, you didn't take enough feathers! In addition to not cutting the feathers back far enough, she's probably still getting lift in part because you clipped all of the feathers on the wing. You really only need to cut the first 5-8 (maybe 10 for a really strong flier) feathers; fewer for heavier breeds and more for lighter breeds like EE and leghorns. If you have someone who can help you do wings, I prefer to cut each feather individually using a pair of dog nail clippers. I spread the wing while someone else holds (putting a towel over their head will help make them calmer if they are struggling a bit) and, looking at the underside of the wing, clip each feather shaft at the base (or, in the case of blood feathers, where the feather has come out of the sheath already and there is no blood supply). This looks much neater and you don't have to worry about accidentally cutting a blood feather the way you do with blindly cutting all the feathers at once across the top side.
