Being of bad character is not generally probative of guilt of a specific offence, woofgang, and so evidence of it is not admitted. That the accused has previous for like offences might be probative, obviously so in 'mark of Zorro' cases where the present offence has unusual characteristics which the previous ones shared but also if he claimed to be of good character or had attacked the character of a prosecution witness; in those two cases his character is brought in to help the jury to decide who should be believed
Yes, it would be open to the prosecution to call witnesses to rebut a defence character witness but it would only happen when that witness made false assertions of fact e.g. they asserted that the man had an impeccable army record when , in fact, his army service was blighted by constant disobedience and bad behaviour and he was discharged for it. Otherwise character assessment is personal to the individual., it is opinion, and not subject to rebuttal.