First, let's start with seeing if Rainbow Dash has ADHD and autism. Those are the most common disorders that Rainbow is diagnosed with.

Since Rainbow is an adult, she will need to have at least five of the symptoms for ADHD.

Inattentive presentation:

Fails to give close attention to details or makes careless mistakes. (No)
Has difficulty sustaining attention. (Yes)
Does not appear to listen. (Yes)
Struggles to follow through on instructions. (No)
Has difficulty with organization. (No)
Avoids or dislikes tasks requiring a lot of thinking. (Yes)
Loses things. (No)
Is easily distracted. (Yes)
Is forgetful in daily activities. (No)

Hyperactive-impulsive presentation:

Fidgets with hands or feet or squirms in chair. (Sometimes)
Has difficulty remaining seated. (Yes)
Runs about or climbs excessively in children; extreme restlessness in adults. (Yes)
Difficulty engaging in activities quietly. (No)
Acts as if driven by a motor; adults will often feel inside like they were driven by a motor. (Yes)
Talks excessively. (Not excessively)
Blurts out answers before questions have been completed. (No)
Difficulty waiting or taking turns. (Yes)
Interrupts or intrudes upon others. (Sometimes)

Combined inattentive & hyperactive-impulsive presentation:

Has symptoms from both of the above presentations. (She does)

She does show to have ADHD. However, did Rainbow have these symptoms when she was twelve? If we assume that ponies get their cutie marks at around the same age they reach puberty, we can look at the events of when Rainbow got her cutie mark. She showed to be focused, she didn't show to have any distractions, she was able to focus on the task at hand (hoof) and win the race. In fact, she was energetic ,but she didn't show to have ADHD.

Does Rainbow have ADHD? Unknown. She may have ADHD, but from the little information we now of her as a filly, she doesn't have it.

She's energetic and has difficulty paying attention to things she has no interest in. That's just the type of pony she is.

What about Autism?

A: Persistent Deficits in Social Communication and Social interaction, not accounted for by general developmental delays, and manifested by 3 of 3 symptoms.
A1. Deficits in social‐emotional reciprocity; ranging from abnormal social approach and failure of normal back and forth conversation through reduced sharing of interests, emotions, and affect and response to total lack of initiation of social interaction.
Abnormal Social Approach (No)
Failure of normal back and forth conversation (No)
Reduced sharing of interests (No)
Reduced Sharing of Emotions/affect (No)
Lack of iniciation of social interaction (No)
Poor social imitation (No)

A2. Deficits in nonverbal communicative behaviors used for social interaction; ranging from poorly integrated‐verbal and nonverbal communication, through abnormalities in eye contact and body‐language, or deficits in understanding and use of nonverbal communication, to total lack of facial expression or gestures

Impairments in social use of eye contact (No)
Impairments in the use and understanding of body postures (No)
Impairment in the use and understanding of gestures (No)
Abnormal volume, pitch, intonation, rate, rhythym, stress, prosody, or volume in speech (No)
Abnormalities in use of understanding of affect (No)
Lack of coordinated verbal and nonverbal communication (No)
Lack of coordinated non-verbal communications (No)

A3. Deficits in developing and maintaining relationships, appropriate to developmental level (beyond those with caregivers); ranging from difficulties adjusting behavior to suit different social contexts through difficulties in sharing imaginative play and in making friends to an apparent absence of interest in people.

Dificits in developing and maintaning relationships, appropriate to developmental level (No)
Difficulties adjusting behavior to suit social context (No)
Difficulties in sharing imaginative play (No)
Difficulty in making friends (No)
Absence of interest in others (No)

She doesn't fit any of the A symptoms. 0/3

B. Restricted, Repetitive patterns, behavior, interests, or activities as manifested by at least 2 of 4 symptoms:
B1: Stereotyped or repetitive speech, motor movements, or objects; (such as a simple motor stereotypes, echolalia, repetitive use of objects, or idiosyncratic phrases).
Stereotyped or repetitive speech (No)
Stereotyped or repetitive motor movements (No)
Stereotype or repetitive use of objects (No)

B2. Excessive adherence to routines, ritualied patterns of verbal or nonverbal behavior, or excessive resistance to change; (such as motoric rituals, instance on same rout or food, repetitive questioning, or extreme distress at small changes.)
Adherence to routine (No)
Ritualized patterns of verbal and nonverbal behavior (No)
Excessive resistance to change (No)
Rigit thinking (No)

B3. Highly restricted, fixated interests that are abnormal in intensity or focus; (such as a strong attatchment to or preoccupation with unusual objects, excessive circumscribed or perseverative interests).
Preoccupations; obsessions (She loves the wonderbolts, but she's not obsessed with them. (No)
Interests that are abnormal in intensity (No)
Narrow range of interests (No)
Focus on the same few objects, topics, or activities (She focuses a lot on flying, but that's because she wants to be a Wonderbolt and trains for it. So, no)
Preoccupation with numbers, letters, symbols (No)
Being overly perfectionistic (No)
Excessive focus on nonrelatant or nonfuncitonal parts of objects (No)
Preoccupations (No)
Attatchment to unusual inanimate object (No)
Having to carr around or hold specific or unusual objects (No)
Unusual fears (No)

B4. Hyper-or hypo-reactivity to sensory input or unusual interest in sensory aspects of environment; (such as apparent indifference to pain/heat/cold, adverse response to specific sounds or textures, excessive smelling or touching of objects, facination with lights or spinning obects).
High tolerance for pain (Not unusually high)
Poking own eyes (No)
Preoccupation with texture or touch (includes attraction/averstion to texture (No)
Unusual visual exploration/activity (No)
Unusual sensory exploration with objects (No)
She doesn't fit any of the B symptoms: 0/4
C. Symptoms must be present in early childhood (but may not become fully manifest until social demands exceed limited capacities)
No

D. Symptoms together limit and impair everyday funcitoning.
She doesn't have symptoms.

Diagnosis: Not on the autistic spectrum. Come one! Having an interest doesn't automatically make someone autistic!

Narcisistic Personality Disorder?

Having an exaggerated sense of self-importance (She actually is important)

Expecting to be recognized as superior even without achievements that warrant it (No)

Exaggerating your achievements and talents (There's no exaggerating her achievements or talents. She brags a lot, but there's no exaggeration)

Being preoccupied with fantasies about success, power, brilliance, beauty or the perfect mate (No)

Believing that you are superior and can only be understood by or associate with equally special people (No)

Requiring constant admiration (She loves it, but she doesn't require it)

Having a sense of entitlement (No)

Expecting special favors and unquestioning compliance with your expectations (No)

Taking advantage of others to get what you want (No)

Having an inability or unwillingness to recognize the needs and feelings of others (No)

Being envious of others and believing others envy you (No)

Behaving in an arrogant or haughty manner (She is arrogant. I won't deny that.)

Having only one symptoms doesn't qualify for a diagnosis. Rainbow Dash is not naricistic! Neither is Rarity, but I'll get to her later.

What does she have?

Bipolar Disorder? No

Schizophrenia? No

Borderline Personality Disorder? No

Dyslexia? No (Rainbow showed no difficulty reading in "Read it and Weep" so I'm going with no)

Anxiety? No. (She gets anxious, but she doesn't have anxiety)

Depression? No.

In fact, Rainbow Dash doesn't have any mental illness. I could imagine her being wrongfully diagnosed with ADHD and put on medications because of her high energy levels, but they don't interfere with her ability to function.

Diagnosis: Rainbow Dash is neurotypical.