Thanks for the suggestions.
Regarding the different icon, the embedded calendar and event list will display whatever icon you've uploaded for that event. When you edit an event in Elvanto you can click on the icon above the 'Previous/Next' buttons, and give it it's own icon there. That way it will show the same icon on the embedded list :)
For the Previous/Next month in the events list, this is something that's been requested before and I'll add your support to.
I'll put a feature request in for the ability to add the Organizer's details to an embedded calendar for you. Most likely this will be a per-event settings.
For now, to help with number 2 I would suggest looking into our APIs calendar/events/getAll endpoint.
Hope this helps