It's unclear, but possible.
At the start of Volume 1 when she attempts to leave her bed, she immediately collapses:
She then had to take out Buck while on the floor, before using a wheelchair to steal the Pussy Wagon. At this point we are shown a 10 minute sequence about O-Ren Ishii, before The Bride finally regains control of her legs. The next shot is clearly daytime, so we can presume it took her all night to overcome this.
It's unlikely this was caused by muscle atrophy which is a symptom of coma, as she had no problem stabbing or slamming Budd's head in the door, nor dragging herself around or operating the wheelchair which indicates her arms are working fine. With this in mind, I believe this is a psychological injury caused by her trauma which could be labelled brain damage.
As to your examples, I disagree that these are symptoms of brain damage, but simply the result of two highly trained professionals fighting for their lives, which is necessarily going to be brutal and dirty:
Despite her previous reputation as the best female fighter in the world, the Bride does not look as a better fighter in her encounters with Vernita
The defining feature of the fight with Vernita is the location: a living room. The Bride quickly takes the upper hand, but at every point one of them is knocked down, they immediately grab something to hand to defend themselves with (the leg of a broken table, a fire poker, a frying pan, a kitchen knife). This is not the sort of fight which demonstrates real fighting skills, it's dirty.
O-Ren has been killing people with swords since she was eleven years old. We don't know that much about The Bride's childhood, but I think it's a fair assumption O-Ren started much younger than her and definitely had the upper hand. It may be that the only reason The Bride won was due to her training with Pai Mei, which O-Ren didn't receive.
Elle, like The Bride, was taught by Pai Mei, and they were very closely matched. Arguably the only reason she lost, was that Pai Mei took her other eye, which allowed The Bride to instantly blind her with a single hand movement. Presumably, Pai Mei taught The Bride this, but not Elle for obvious reasons.
Also she is outsmarted by Budd.
Outsmarted is a bit strong. The Bride didn't know that Budd knew she was coming, so she had no reason to expect him to shoot her through a closed door.
Also she fails to anticipate Bill's attack.
This is certainly questionable judgement seeing as he attempted to murder her, but he was very convincing, and the joy at being reunited with a child she assumed dead must be an indescribable experience.
So, is the Bride just as efficient fighter as she was before her head injury?
I believe so. The most important point is that they are the Deadly Viper Assassination Squad. They are all excellent fighters, but their primary task was killing people who weren't expecting it, which is very different from killing someone who has all the same training as you and is expecting you to come.