Ŝi sentas sin trankvila.

In Esperanto, senti sin + adjective is the normal pattern for saying that someone feels a certain way.

So Ŝi sentas sin trankvila is literally something like She feels herself calm. English usually leaves out herself, but Esperanto normally keeps the reflexive pronoun in this structure.

Because the object refers back to the subject.

• ŝi = she
• ŝin = her
• sin = herself, himself, itself, themselves, when it refers back to the subject

Since the subject is Ŝi, Esperanto uses the reflexive form sin rather than ŝin.

Because sin is the direct-object form.

Here, sentas takes a direct object, so Esperanto uses sin with -n.

You would use si after a preposition, for example:

• pri si = about herself / himself
• al si = to herself / himself

Because it describes the person, not the manner of the action.

• trankvila = calm, describing someone or something
• trankvile = calmly, describing how an action is done

So here, the idea is that she is in a calm state, not that she is doing the feeling calmly.

A good comparison:

• Ŝi sentas sin trankvila = she feels calm
• Ŝi parolas trankvile = she speaks calmly

Because trankvila is not the direct object. The direct object is sin.

trankvila is a predicate complement: it tells you what state the person is in. In this kind of sentence, Esperanto normally leaves that adjective without -n.

So:

• sin = the object
• trankvila = the description of that object

No. Esperanto adjectives do not change for gender.

trankvila can describe:

• li
• ŝi
• ĝi

The adjective only changes for things like number:

• Ŝi sentas sin trankvila
• Ili sentas sin trankvilaj

So -a does not mean feminine.

No. si/sin is only used for the third person.

So you say:

• Mi sentas min trankvila
• Vi sentas vin trankvila
• Ŝi sentas sin trankvila

That is a very common rule for learners to remember.

Not exactly.

• Ŝi estas trankvila says that she is calm.
• Ŝi sentas sin trankvila says that she feels calm.

The first is a plain statement about her state. The second emphasizes her own experience or perception of that state.

It marks the present tense.

So:

• sentas = feels / is feeling
• sentis = felt
• sentos = will feel

This is one of the basic Esperanto verb endings:

• -as present
• -is past
• -os future