I see that tags are present only for questions. How can I get the answers to these questions given in last week (if any) having specific tags?
I'd still use the /questions method, and change the filter to include answer details. E.g. this query lists all Stack Apps answers in the support tag, newest first. All I had to do was to edit the default filter and check the checkboxes for the fields 'answer' → 'body' and 'question' → 'answers'.
There is no way to filter on answer date upfront, but if you sort by question activity, you can ignore older answers and stop searching when you see a question with a
last_activity_date older than a week ago.