Is it permissible to speak during the Friday call to prayer (adhan) before the sermon (khutbah), and would the Friday prayer then be valid?

Speaking on Friday is forbidden during the sermon, not before or after it. So, when the Imam stands to deliver the sermon, it is obligatory to stop speaking and listen to him. As for speaking during the Athan (call to prayer), there is no sin in it, but it is better to busy oneself with repeating the Athan due to the Prophet's (peace be upon him) saying: "When you hear the call (to prayer), say as the Mu'aththin says."