Η φίλη μου δεν θυμώνει εύκολα, γελάει δυνατά και βοηθάει τον εαυτό της να μείνει ήρεμη.

In Greek, the usual way to say “my X” is:

ο/η/το + noun + μου

So η φίλη μου literally is “the friend my,” and that’s the normal, neutral pattern.
You can sometimes drop the article (just φίλη μου), but that sounds more emotional, poetic, or vocative (e.g. calling out to someone: Φίλη μου! “My friend!”).
In a simple descriptive sentence like this, η φίλη μου is the standard form.

μου is an unstressed possessive pronoun meaning “my”. In modern Greek it usually:

• comes after the noun: η φίλη μου = my friend
• is unstressed (a “clitic”), so it cannot stand alone and doesn’t take its own accent

When you want to say “my + noun,” you don’t put μου before the noun like in English; you attach it after the noun (with the definite article before the noun):
το βιβλίο μου (my book), ο αδερφός μου (my brother), η φίλη μου (my friend).

The verb θυμώνω means “to get angry / to become angry” (or “to be angry” in some contexts).
Here δεν θυμώνει is 3rd person singular, present tense, indicative: “she doesn’t get angry / she doesn’t tend to get angry.”

• δεν θυμώνει = “she doesn’t get angry” (generally, habitually)
• To say “she got angry (once)” you’d use the past aorist: θύμωσε → δεν θύμωσε = “she didn’t get angry.”
• To say “she is angry (right now)” you would more often use είναι θυμωμένη (“she is angry,” literally “she is angered”), using an adjective.

So the sentence means she is not easily getting angry as a general character trait.

In Greek, the basic rule is:

δεν (or μη(ν) in some structures) goes immediately before the verb it negates.

So:

• Δεν θυμώνει – she doesn’t get angry
• Δεν γελάει – she doesn’t laugh
• Δεν βοηθάει – she doesn’t help

You generally cannot move δεν somewhere else in the sentence the way you sometimes can with “not” in English.

εύκολα is an adverb meaning “easily.” It is formed from the adjective εύκολος (“easy”).

A very common pattern in Greek is:

• adjective in -ος → adverb in -α
  
  • καλός → καλά (good → well)
  • δυνατός → δυνατά (strong → strongly / loudly)
  • εύκολος → εύκολα (easy → easily)

Position-wise, εύκολα usually comes after the verb:
Δεν θυμώνει εύκολα = She doesn’t get angry easily.
You could also say Εύκολα δεν θυμώνει, but that sounds more marked/emphatic (roughly: “She certainly doesn’t get angry easily”).

Yes. γελάει and γελά are both common modern spellings/pronunciations for “(she) laughs.”

The verb is γελάω (or γελώ) = “to laugh.” In the present tense:

• (εγώ) γελάω / γελώ – I laugh
• (εσύ) γελάς – you laugh
• (αυτός/αυτή) γελάει / γελά – he/she laughs

In everyday modern Greek, verbs in -άω often have two parallel forms in the 3rd person: -άει and -ά. Both γελάει and γελά are correct and pronounced almost the same in speech.

In this sentence δυνατά means “loudly”:
γελάει δυνατά = “she laughs loudly.”

δυνατός is an adjective that means both “strong” and “loud” (for sounds). Its adverb δυνατά can therefore mean:

• strongly / with force – e.g. χτυπάει δυνατά = he/she hits hard
• loudly (about sounds or voice) – e.g. μιλάει δυνατά = he/she speaks loudly

Here, with γελάει, the natural interpretation is “loudly.”

Modern Greek generally does not form reflexive verbs in the same systematic way as some other European languages (like German sich helfen or Spanish ayudarse).
Instead, it often uses the noun εαυτός (“self”) with a possessive pronoun:

βοηθάω + τον εαυτό μου/σου/του/της/μας/σας/τους

So:

• βοηθάει τον εαυτό της = “she helps herself”
• βοηθάω τον εαυτό μου = I help myself

There is a passive form βοηθιέμαι, but it means “to be helped” (by someone), not “to help oneself.” So for “help oneself,” the natural pattern is βοηθάω τον εαυτό μου / σου / της, not a reflexive verb form.

Because εαυτός is a masculine noun in Greek, regardless of who it refers to.

• ο εαυτός = the self (grammatically masculine)

So in the accusative singular you must use the masculine article:

• τον εαυτό μου – myself
• τον εαυτό σου – yourself
• τον εαυτό του – himself
• τον εαυτό της – herself
• τον εαυτό μας, σας, τους – ourselves, etc.

The grammatical gender of εαυτός (masculine) controls the article τον, not the biological gender of the person. The της at the end (“her”) shows that it is her self.

να μείνει is the subjunctive form of the verb μένω (“to stay, to remain”) in the aorist:

• να μένει – present subjunctive
• να μείνει – aorist subjunctive

After verbs like βοηθάει (“she helps”), Greek commonly uses να + subjunctive:

• βοηθάει τον εαυτό της να μείνει ήρεμη = she helps herself (so that she) stay/remain calm.

Using να μείνει focuses on the result (to be/remain calm as a whole), often seen as a single situation.
να μένει would emphasize an ongoing, repeated or continuous staying calm. In many contexts both are possible, but να μείνει ήρεμη sounds very natural here.

ήρεμη is a feminine adjective meaning “calm”, agreeing with the subject η φίλη μου (feminine, singular).

The verb μένω here behaves like “to be / to remain,” so it takes a predicate adjective, not an adverb:

• μένει ήρεμη = she remains calm (she is calm)
• ήρεμη agrees in gender, number and case with η φίλη (μου).

ήρεμα is the adverb “calmly,” which would describe how she does an action (e.g. μιλάει ήρεμα = she speaks calmly).
Here the sentence is about her state (to stay calm), so the adjective ήρεμη is required.

Yes, Greek allows some flexibility here.

1. Comma and “και”
   The original:
   Η φίλη μου δεν θυμώνει εύκολα, γελάει δυνατά και βοηθάει τον εαυτό της να μείνει ήρεμη.
   You could also say:
   Η φίλη μου δεν θυμώνει εύκολα, γελάει δυνατά, και βοηθάει τον εαυτό της να μείνει ήρεμη.
   The extra comma before και is optional and mainly a stylistic choice.

2. Repeating the subject
   You can repeat η φίλη μου for emphasis or clarity, but it’s not necessary:

   • Η φίλη μου δεν θυμώνει εύκολα, η φίλη μου γελάει δυνατά και βοηθάει…
     This sounds more emphatic or rhetorical; normally you state the subject once and then just list the verbs.

3. Order of the three actions
   You could, in principle, reorder the three verbs, but then you change the emphasis or the flow. The given order is natural and smooth in Greek.