Reconocer mi debilidad es parte de cuidar mi salud mental.

Spanish often uses the infinitive as a noun, especially as the subject of a sentence.
Reconocer mi debilidad functions as a single idea, like Recognizing my weakness in English. You cannot normally use the gerund (reconociendo) as the subject, so the infinitive is the natural choice.

The subject is understood to be yo (I), even though it is not said. Spanish usually drops subject pronouns when the context makes them clear. Here, Reconocer mi debilidad is something that I do, but yo is omitted because it is obvious.

With parte, Spanish almost always uses the preposition de to mean part of: es parte de algo.
In this sentence, cuidar mi salud mental is treated like a noun phrase (the act of taking care of my mental health), so you say parte de cuidar, not parte para cuidar.

Again, the infinitive is used as a noun-like form: cuidar mi salud mental = taking care of my mental health.
The gerund cuidando is used for ongoing actions (for example, estoy cuidando), but not normally as a subject or object in this way. So de cuidar mi salud mental is the correct structure.

In Latin American Spanish, cuidar is most often used without de when it has a direct object: cuidar a alguien, cuidar algo.
Cuidar de exists, but it is more common in Spain and often sounds a bit more formal or old-fashioned in Latin America. In most Latin American contexts, cuidar mi salud mental is the most natural version.

Spanish does not combine a definite article and a possessive before the same noun. You either say la salud mental (the mental health, in general) or mi salud mental (my mental health), but not la mi salud mental.
With possessives like mi, tu, su, you normally do not use an article.

Yes, Reconocer mis debilidades is grammatically correct.
Mi debilidad (singular) sounds like a general, overall weakness or a particular main weakness. Mis debilidades (plural) emphasizes that you have several specific weaknesses. Both are possible; the nuance is slightly different.

Mi debilidad makes it personal: my weakness.
La debilidad would sound more like weakness in general or the weakness (of something previously mentioned), not specifically yours. In a personal, reflective sentence like this, the possessive mi is the natural choice.

Yes, forma parte de is very common and fully correct:
Reconocer mi debilidad forma parte de cuidar mi salud mental.
The meaning is practically the same as es parte de; forma parte de can sound slightly more formal or more clearly emphasize that it is one component of a larger process.

No reflexive pronoun is needed in this sentence because the direct objects are already clear: mi debilidad and mi salud mental.
Reconocerme would mean to recognize myself (as in I saw myself in the mirror and recognized myself), which is not the idea here.
Similarly, cuidarme would mean to take care of myself in a broader sense, but here the focus is specifically on my mental health, so cuidar mi salud mental is more precise.