All dogs have blood in their nails, vein is called the Wick (that's what I recall). Red is unusual to my experience because the blood is seen through the semi-transparent nail material. I see the wick more as black, but it could under bright light appear red, I suppose. This is normal, and if the nail is trimmed too short, cuts the wick the nail will bleed. This can be stopped or can stop on its own.
Unless there is more to the nail issue, I'd not worry about it. Water is important, and if our dog is old recovery can take longer.
Let her rest for a while. It's probably the antibiotic that's making her drowsy. Make her drink water to avoid dehydration. You can use a syringe to force the water down her throat, but be sure to be very gentle to avoid injuring her.