Where is the reflexive? Is it not needed here? This sounds very odd, as in English, it should be, "I look around myself."
I'm not seeing where a reflexive would be necessary. One might say Mi rigardas ĉirkaŭ mi mem. But that would be, as you suggest, I look around myself. which might be, "I'm checking my clothes and the ground in the immediate area to see where my keys have gone." But "I look around me." sounds as if someone sits up, comes to a high point or otherwise has an awareness that looking about the whole area is a good thing. "I woke up and looked around me." Mi vekis kaj rigardis ĉirkaŭ mi. "At the mountain's peak I stood, and looked around me." Ĉe la monta pinto mi staris, kaj rigardis ĉirkaŭ mi. "I walked into the room, etc." Mi promenis en la ĉambron, ktp, y así sucesivamente.
Now, if it were in the second or third person…
It generally means something along the lines of I look at the things that are near to me in all directions but can mean things further away, or even things that are on your person - typically it is said if you have lost something, or find yourself somewhere new and strange and you are getting acquainted with your environment, but again that's not exclusive!
Ah, mi neniu diras la anglan frazon, do mi ne komprenis gxin. Mi direblas "I look around myself", sed mi konfuzis.
Could I say "mi rigardas cxirkau" to just say "I look around" (without specifying "around me/sth. else)? Cheers!
Are all the subject and object pronouns are the same for each 'person'? I had never noticed before if so.
All direct objects of verbs end in -n, including pronouns. But not if it's the object of a preposition unless there's movement.
I love him: Mi amas lin.
He loves me: Li amas min.
I give the letter to him: Mi donas la leteron al li. (not al lin)
I jump on the table: Mi saltas sur la tablo. (I am standing on the table and jumping up and down.)
I jump onto the table: Mi saltas sur la tablon. (I am standing somewhere else, I jump and land on the table.)
Usually, the object pronouns end in ~n same as other nouns. That is one of the basic rules from Zamenhof. However, this sentence is an example of the Rule: All prepositions govern the nominative case.
Ĉirkaŭ is a preposition (yes, it is, don't argue with me) and when a (pro)noun follows a preposition (after a verb), unless direction needs to be specified, it falls back into the subjective (or nominative) form.
Why wouldn't it be "Mi rigardas cxirkaux si?" because the subject is already stated, and is the same as the object.