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The behaviors that you've described are not really indicative of ADHD for which Ritalin is at times prescribed.
These seem to be more of personality traits.
Is she the only child?
If you've noticed the ability for her to control herself at school and act differently at home, you're right- she's exercising some control over her behavior( it seems to be volitional).
Her behavior should have not gotten to the point where her father's health was affected to that extent.
Getting her an evaluation by a psychologist would be appropriate and something to look into further. She can receive therapy. Only certain conditions are treated with medication while others are treated mainly w/ therapy.
The fact that you've mentioned that she does not feel or show empathy is a red flag. It is found in those individuals with narcissistic traits. It is one of the main sympltoms along with the need for attention/approval/admiration.
In therapy, she can possibly learn coping and social skills. The former to manage her feelings when frustrated or feels ignored, the latter in how the channel these appropriately.
The fact that she has a strong sense of what she believes can be directed to positive even altruistic behaviors i.e. volunteering (helping others) and gaining a sense of pride through her own contribution. That would also allow her to foser a healthy sense of self. Because, acting out powerful around others (or bullying them if she ever does), can be a sign of underlined low self esteem she's trying to compensate for.
According to the STEP (systematic training for effective parenting), children can misbehave because of 1) sense of inadequacy, 2) attention 3) power struggle, 4) revenge
The direction you were inquiring about would be 1) have her be properly evaluated 2) discuss treatment options w/ the clinician 3) set a weekly family time where every family members discusses how their week had gone so far (including how other family members may be affecting them) 4) try to speak w/ her or encourage her to journal her feelings (that could foster or awaken empathy ) 5) help her determine what volunteering she would like to try (even if it is going w/ you once a week to an animal shelter to feed the animals or a nursing home to read to the elderly)
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At this age, she is still learning about herself through her interactions w/ others. You may always stress to her by giving her some specific examples of how asserting her way can be used positively as well as negatively. Cases where she's taken up for someone else (defended or helped another) or when she's remained strong headed just to get her way, can be some concrete examples to talk with her about. You would want to help her identify what she felt in each situation (pride, anger, assertiveness, helpful, etc), and point out to her how the positive feelings and positive outcomes are within her control.
In order to learn social skills, she'd have to continue to interact w/ her peers. If possible, enrolling her in some sort of a club or sport may be helpful. Even if it is a competitive sport with clearly defined rules i.e. karate, she can learn respect as well as cooperation. If there are any summer camps for children, that too is another thing to look into. Any activity where she would learn and need to practice self control while working w/ others can be helpful. Some of these activities can center on the need to cooperate w/ peers in order to achieve something as a team rather than individually.
You could also devise a system where she earns some sort of a desirable outcome for positive behavior and a negative one for inappropriate behavior. For instance, you notice her interacting in a helpful/patient manner w/ another. You reward her w/ something that is pleasurable to her i.e. extra play time, going to bed later, etc. If the opposite occurs, you use a consequence. Of course, you'd want to let her know ahead of time that this is what you'd be doing and let her know what the expectations are. This would have to be reinforced without you giving up the first few times if she resists. And, she may just as a form of control.
It would be helpful for her sister not to feed into her behavior. The age difference is large and her older sister does not have to antagonize her. Her sister can be someone that you ask to assist with this behavior modification of sorts by also rewarding her positive behaviors. If she's acting well with her, then her older sister may give her something or take her for a treat. Encouragement and praise of positive behavior is a reward in itself (complimenting her when she does well).
If you could find a movie for children where one of the characters is being bullied by others, and discuss it afterwords, it would help her process her feelings (whether she's feeling empathy, anger, indifference, etc). She may need to feel a sense of belonging. If she does not, she could be acting the opposite way (defense mechanism- reaction formation) Instead of attracting others to herself, she pushes them away in order not to be rejected by them first.