Mi sentas min plej trankvila hejme.

Because min is the accusative form of mi. After sentas, Esperanto often uses an object when you mean I feel myself ....

So:

• Mi sentas min laca. = I feel tired.
• Mi sentas min plej trankvila hejme. = I feel calmest at home.

Using mi there would make it a second subject, which is not what the sentence needs.

Because sin is only for the third person.

So Esperanto uses:

• mi ... min
• vi ... vin
• li/ŝi/ĝi/ili ... sin

That means:

• Mi sentas min ... = correct
• Li sentas sin ... = correct
• Mi sentas sin ... = incorrect

Because trankvila describes me, not the way the action happens.

In this sentence, trankvila means calm as a state: I feel calm.

• trankvila = calm, peaceful
• trankvile = calmly, in a calm way

Compare:

• Mi sentas min trankvila. = I feel calm.
• Mi parolas trankvile. = I speak calmly.

An English speaker may want an adverb because English often uses forms like I feel good without obvious case marking, but Esperanto keeps the adjective here.

Because it is a predicative adjective, not a direct object.

The direct object is already min. The adjective trankvila tells what state that object is in. In Esperanto, this kind of adjective normally does not take the accusative ending.

So:

• Mi sentas min laca.
• Mi trovas lin interesa.
• Ili elektis ŝin prezidanto.

The -n goes on the object pronoun or noun, not on the describing word after it.

Plej means most. It gives the superlative degree.

• trankvila = calm
• pli trankvila = calmer
• plej trankvila = calmest / most calm

So the sentence says that among different places or situations, home is where the speaker feels the most calm.

If you used pli, you would need something to compare it with:

• Mi sentas min pli trankvila hejme ol en la laborejo.
  = I feel calmer at home than at work.

This is something learners often notice because many textbook superlatives use la plej.

In a sentence like this, plej can work naturally to show the highest degree in context: I feel most calm at home. Some speakers and textbooks may prefer la plej trankvila in more explicit superlative phrasing.

So the safest thing to learn is:

• plej marks the highest degree
• la plej is also very common, especially in clear noun-based superlatives like la plej alta domo

In this sentence, the main point is that plej shows the strongest degree of calm.

Hejme means at home.

It comes from hejmo = home, with the adverb ending -e:

• hejmo = home
• hejme = at home
• hejmen = homeward, to home

So hejme gives a location, not movement.

Because the sentence talks about location, not motion.

• hejme = at home
• hejmen = to home, homeward

Compare:

• Mi sentas min plej trankvila hejme. = I feel calmest at home.
• Mi iras hejmen. = I am going home.

The -n on place words often shows direction toward something.

Because Esperanto normally does not need esti in this pattern.

The structure senti + object + adjective is already enough:

• Mi sentas min laca.
• Ŝi sentas sin feliĉa.
• Ni sentas nin sekuraj.

This is the normal way to say feel tired / feel happy / feel safe. Adding esti would sound unnecessary or awkward in ordinary Esperanto.

Yes, very common.

It is one of the basic ways Esperanto expresses personal physical or emotional state.

Examples:

• Mi sentas min malsana. = I feel sick.
• Li sentas sin sola. = He feels lonely.
• Ni sentas nin pretaj. = We feel ready.

So Mi sentas min plej trankvila hejme is built on a very normal pattern.

Not completely. Esperanto word order is fairly flexible, because the endings show the grammatical roles.

The original sentence is neutral and natural:

• Mi sentas min plej trankvila hejme.

But you could also move hejme for emphasis:

• Hejme mi sentas min plej trankvila.

That puts more focus on at home. The meaning stays basically the same.

You could, but hejme is much more natural for the ordinary idea of at home.

• hejme = at home, in the usual idiomatic sense
• en la hejmo = in the home / inside the home

En la hejmo sounds more literal and physical, as if you are emphasizing the building itself. In most everyday situations, Esperanto prefers hejme.