Is it possible to use the Marketo web api to send an email without that email being a part of a Request Campaign?
Request Campaign and Schedule Campaign are the two API endpoints designed specifically to trigger campaigns without any other conditions or dependencies (and with the added feature of temporary tokens).
You can also create a Trigger Smart Campaign based on any number of criteria, including activities such as a lead field being updated via API.
Is it possible to use the Marketo web api to add an attachment to an email and then send that email?
No. Marketo does not support attachments, for good reason.