Although the words you listed are all semantically related, they each have quite distinct grammatical roles in the typical usage (with little overlapping) that you never want to miss.
As function words: 自分/己 vs 自身/自ら vs 自己 (vs 自)
自分 is a standalone pronoun that substitutes the same referent (noun) that appears in the context. While it certainly can be categorized as "reflexive pronoun", its chief use cases are sentences which have multiple actions that involve the same participant. Don't expect too much for it to really translate oneself or such in English, and vice versa. Most of the times they have better lexical expressions; try to look up those parallel corpora.
He does not forgive those who laughed at him (*himself).
He washed himself.
It also represents various meanings usually translated with adverbs in English:
自分で by oneself; personally; alone
自分から on its own will; spontaneously
Sometimes, 自分 appears without an apparent antecedent. In this case, it means general "you". Japanese usually omits such a pronoun, but it is required, like above, when it has references to the same agent. (Natural translation may become "me/myself" or "him(self)" and such according to the context.)
視点を持て Have a perspective.
自分の視点を持て Have your own perspective.
As I said "standalone", 自分 does not form compounds with other nouns. It is only accompanied by particles or affixes (affixable words).
自分たち oneselves (plural)
自分らしさ one's own style; being oneself
自分どうし (see what it means)
己【おのれ】 is simply an archaic and/or solemn variant of 自分.
Meanwhile, 自身 is a suffix, always attached to another noun to add meanings like the very —; — oneself; — per se etc. The word must be put right after the noun, though the resulted translation may not need to be. Sometimes it is just for extra emphasis; other times not:
The boss is fragile in itself, but you can't damage it before shutting down the shield.
If you combine 自分 with 自身, you get 自分自身.
Note that both 自分 and 自身 are mainly for animate nouns. In most cases you should use 自体 instead of 自身 for the "per se" sense. There is no exact replacement of 自分 for inanimate nouns, so we usually use それ or repeating the noun in such context.
Banknotes do not have value on their own.
自【みずか】ら is a synonym of the suffix 自身 with a couple of reservations: (1) there is no 自分 + 自ら; (2) usable with animate and inanimate alike.
自己 is almost always used as prefix to make compounds that means self- or auto-. It has a large number of derivatives:
自己修復材料 self-healing materials
Lastly, there is another prefix 自-, which is either (a) same with 自己 exclusively for building two- or four-kanji idioms; (b) of one's; or (c) your own group, "friend" as opposed to "foe", or "we" to "they".
自業自得 You reap what you sow.
自動詞 intransitive verb
自チーム one's team
自陣 own base, own half, home territory
自分 is, besides the above, used for true noun for (one's) self. It is the simplest word that kids would use for this sense. Has some compounds:
自分勝手 selfish, self-righteous
自分探し finding yourself
自身 can also be used apparently alone in bookish styles, which technically should be taken as omission of the previous noun. Not as truly independent to form compounds. However, compared to 自分, it focuses more on external properties, such as: one's past, one's life, one's deed, one's belonging etc (=自分の身, 我が身). Also applies to 自ら.
Bolt broke his own world record.
自己 is rarely used as noun except for self as a psychological term. Otherwise you can safely forget about it in modern writing.
自力 is literally "one's own capacity". Almost only used in the collocation 自力で "by oneself (without others' help)" = 自分（の力）で.
自力救済 self-help (legal term)
自力更生 self-reliance (Chinese slogan, lit. "revitalize on our own")
As personal pronouns
自分 is sometimes a first-person pronoun I, but also dialectal informal you in Kansai.
己 is also informal you in some (mostly Western) dialects.
自ら can be used alone as an adverb: with one's own hands/initiative; at first hand.
Some situations can be expressed with multiple options.
The CEO supervise the project in person.
= 社長が自分で計画の音頭を取る。 (pronoun + case)
= 社長自身が計画の音頭を取る。 (suffix)
= 社長が自身で計画の音頭を取る。 ((pseudo-)noun + case)
= 社長自らが計画の音頭を取る。 (suffix)
= 社長が自ら計画の音頭を取る。 (adverb)
× 社長自分が計画の音頭を取る。 (not affix)
× 社長が自分計画の音頭を取る。 (not bare or as affix)
× 社長が自身計画の音頭を取る。 (not bare or as affix)
- 自我: ego (of psychology); note that non-scientific uses are like "alter ego": 別人格, "egoist": 自己中心的 etc.
- 本人: the person him/herself, the principal person, the real McCoy
- 我【われ】: contrary to common assumption, this word is more like "oneself" than "I": 我に返る "come to oneself", 我を忘れる "lose oneself"; in genitive: 我【わ】が家 "one's own home", 我が道 "one's own way"
- 自【おのずか】ら: (of inanimate) spontaneously; as a natural consequence