Is it possible to attach time entries to an existing draft invoice? Will that change the invoice amount if the entries are billable?
Yes. You can attach approved time entries to an invoice, draft or final, by using the Progress Billing feature of Core. Check out this article for details: How to attach time entries to an existing invoice. Note that this won't change the invoice amount, but the time entries will be written up or down to match the original invoice amount.
Note: You should attach time entries in this manner only when you are pre-billing a fixed project. If it is an hourly invoice, you should rather delete the draft and recreate the invoice.