Etymology
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be (v.)

Old English beon, beom, bion "be, exist, come to be, become, happen," from Proto-Germanic *biju- "I am, I will be." This "b-root" is from PIE root *bheue- "to be, exist, grow," and in addition to the words in English it yielded German present first and second person singular (bin, bist, from Old High German bim "I am," bist "thou art"), Latin perfective tenses of esse (fui "I was," etc.), Old Church Slavonic byti "be," Greek phu- "become," Old Irish bi'u "I am," Lithuanian būti "to be," Russian byt' "to be," etc.

The modern verb to be in its entirety represents the merger of two once-distinct verbs, the "b-root" represented by be and the am/was verb, which was itself a conglomerate. Roger Lass ("Old English") describes the verb as "a collection of semantically related paradigm fragments," while Weekley calls it "an accidental conglomeration from the different Old English dial[ect]s." It is the most irregular verb in Modern English and the most common. Collective in all Germanic languages, it has eight different forms in Modern English:

BE (infinitive, subjunctive, imperative); AM (present 1st person singular); ARE (present 2nd person singular and all plural); IS (present 3rd person singular); WAS (past 1st and 3rd persons singular); WERE (past 2nd person singular, all plural; subjunctive); BEING (progressive & present participle; gerund); BEEN (perfect participle).

The paradigm in Old English was: eom, beo (present 1st person singular); eart, bist (present 2nd person singular); is, bið (present 3rd person singular);  sind, sindon, beoð (present plural in all persons); wæs (past 1st and 3rd person singular); wære (past 2nd person singular); wæron (past plural in all persons); wære (singular subjunctive preterit); wæren (plural subjunctive preterit).

The "b-root" had no past tense in Old English, but often served as future tense of am/was. In 13c. it took the place of the infinitive, participle and imperative forms of am/was. Later its plural forms (we beth, ye ben, they be) became standard in Middle English and it made inroads into the singular (I be, thou beest, he beth), but forms of are claimed this turf in the 1500s and replaced be in the plural. For the origin and evolution of the am/was branches of this tangle, see am and was.

That but this blow Might be the be all, and the end all. ["Macbeth" I.vii.5]
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be- 

word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause, seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by;" see by (prep.)). The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).

The Old English prefix also was used to make transitive verbs and as a privative prefix (as in behead). The sense "on all sides, all about" naturally grew to include intensive uses (as in bespatter "spatter about," therefore "spatter very much," besprinkle, etc.). Be- also can be causative, or have just about any sense required. The prefix was productive 16c.-17c. in forming useful words, many of which have not survived, such as bethwack "to thrash soundly" (1550s) and betongue "to assail in speech, to scold" (1630s).

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after (adv., prep.)

Old English æfter "behind; later in time" (adv.); "behind in place; later than in time; in pursuit, following with intent to overtake" (prep.), from of "off" (see off (adv.)) + -ter, a comparative suffix; thus the original meaning was "more away, farther off." Compare Old Norse eptir "after," Old Frisian efter, Dutch achter, Old High German aftar, Gothic aftra "behind;" also see aft. Cognate with Greek apotero "farther off," Old Persian apataram "further."

From c. 1300 as "in imitation of." As a conjunction, "subsequent to the time that," from late Old English. After hours "hours after regular working hours" is from 1814. Afterwit "wisdom that comes too late" is attested from c. 1500 but seems to have fallen from use. After you as an expression in yielding precedence is recorded by 1650.

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after-burner (n.)
1947, "device on the tailpipe of a jet engine to increase thrust," from after + burner.
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be-in (n.)
"a public gathering of hippies" [OED], 1967, from be + in (adv.).
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after-care (n.)
"care given after a course of medical treatment," 1854, from after + care (n.).
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succeed (v.)

late 14c., succeden, intransitive and transitive, "come next after, follow after another; take the place of another, be elected or chosen for" a position, from Old French succeder "to follow on" (14c.) and directly from Latin succedere "come after, follow after; go near to; come under; take the place of," also "go from under, mount up, ascend," hence "get on well, prosper, be victorious," from sub "next to, after" (see sub-) + cedere "go, move" (from PIE root *ked- "to go, yield").

Meaning "to continue, endure" is from early 15c. The sense of "turn out well, have a favorable result" in English is first recorded late 15c., with ellipsis of adverb (succeed well). Of persons, "to be successful," from c. 1500. Related: Succeeded; succeeding.

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sooterkin (n.)
1680s, imaginary rat-like after-birth believed to be gotten by Dutch women by sitting over stoves, 1680s.
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