Event Form for multiple event registrations

Is it possible to create a form which enables people to tick to sign up for a number of different events? (They can either chose one of the events or multiple).

Of course, why not?

You can use a single multi-valued field like Upcoming Registered Events (and Checkboxes field type) on the form. Just make sure to process the form post data promptly, sorting people into Programs based on {Your field name} [contains]. (The next time they post the same form, the value will be overwritten, but since you're already transformed the values sub-into Program memberships, that's fine.)

How do we then push the sign ups for each event into it's own Salesforce campaign and email the registrant with a confirm that they have signed up?

Concentrate on moving people into Marketo programs. You can sync the Marketo programs to SFDC campaigns using that built-in feature.