A half suspension or full suspension mountain bike is the best choice when biking around the highlands or other rural areas.
Follow roads, trails or bike paths. Bikes can damage nature big time when not on trails etc. Take care of nature and be a good row model for others. If something seems to be difficult and dangerous, then it most likely is.
Keep in mind that in Iceland there are very few specific bike paths or roads. You’ll therefore mostly be riding either on paved roads in varying condition, or on gravel roads.