For your second sentence, the ‘Longman Student Grammar of Spoken and Written English’
(and, I suspect, others) uses the term ‘get-passive’, commenting that it
. . . is rare in all registers, but is occasionally used in
conversation. Only five verbs have a notable frequency with the
get-passive . . . Many of these verbs
have a different emphasis when used with the get-passive rather than
the be-passive. With be, they express a state, such as the state
of 'being married' or 'being involved.' With get, they are more
dynamic, describing the processing of getting into that state . . .
Get-passives are typical only in conversation. The written registers usually use become instead.
Get ready is a phrasal verb and, as such, has a progressive form as in your example. Be ready is not a phrasal verb. It is the verb be + the adjective ready. The progressive form of be is not used to describe a state, and that is why ‘He is being ready’ is ungrammatical. The progressive form of be can, however, be used to describe an activity. ‘He is stupid’ describes a continuing state, but ‘He is being stupid’ describes something he is doing now.