Is it forbidden for one upon whom the sacrifice of an expiatory sheep is obligatory to do what is forbidden for one in a state of Ihram, and is there a specific time for its slaughter?

That which is forbidden to a pilgrim in ihram (definition) is not forbidden to you. Thus, performing tawaf (definition) without ritual purity is valid. According to the Hanafis, expiation can be deferred, meaning it is permissible to delay it as long as one's life is not nearing its end. As for the Shafi'is, they differentiate between one who intentionally does so, in which case expiation is immediately due, and one who does not intend it, in which case expiation can be deferred. Completing tawaf without ablution, while knowing its obligation, necessitates immediate expiation. Some scholars mandate expiations immediately in all cases.