I think you will be satisfied with Steve's quirky weblog post called Be-ing: The Bemusing, Busy Prefix be-. The use of this prefix is much more common than you think and does not apply to one particular category of words, to answer your first question.
“Be-” as a prefix goes back to Old English, apparent in such ancient-sounding words as betwixt, betroth, and bereft.
We see it in so many common verbs we use everyday: begin, behave, become, believe, befriend, belong.
And in common prepositions and adjectives: beneath, beside, below, between, beyond, beloved, bereaved.
This prefix is strikingly versatile and productive. The same post explains:
The prefix be- can act as an intensifier, indicating something is thoroughly or excessively done, as in bewitch, bewilder, bedazzle. It can show a verb is affecting or causing something: bedevil, bedim, befoul.
The prefix be- also expresses position: beside, below, between, beneath, behind. Or that something is covered all over or all around: bejeweled, bespattered, bewhiskered. And be- can indicate creation, beget, begin, become, or removal, the end of existence, as in begone, bereave (to take a loved one, especially in death), bereft (deprived of something) and, of course, behead. The multi-tasking prefix be- can turn an intransitive verb into a transitive one as in bemoan and belie. And be- can also turn nouns and adjectives into verbs: befriend, belittle, becalm.
Consequently, the use of be- is not archaic in itself. Each word will be archaic or not according to its own itinerary. However, forming new words with this prefix is outdated now, as Etymonline states:
The prefix was productive 16c.-17c.
The same site gives an interesting detail:
The Old English be- is the unstressed form of bi "by. The form has remained by- in stressed positions and in some more modern formations (bylaw, bygones, bystander).