score:7
Yes, you can travel between Iceland and other Schengen countries without going through any passport checks. I've done this many, many times.
Note, however, that the airline may require a passport (or other Schengen approved travel document) before they'll let you board the aircraft.
Thus, in practice, you'll usually want to have your passport along for trips to Iceland.
Upvote:2
The main purpose of the Schengen rules is to let EU citizens drive across land borders without having to stop and show papers. The unified visa system is just a side effect of this goal. So there was an abolition of systematic passport controls but not an abolition of the requirement to carry passports and to show them to competent authorities upon demand.
Distinct from that are airline security checks, which apply to domestic flights as well. Airlines want to know whom they carry.
Upvote:3
To answer the question about why Ireland is not part of the Schengen Zone: The UK explicitly opted out of the Schengen Agreement. Ireland shares a land border with the UK (Northern Ireland) and there is strong political will for that border to be a soft border. As a result, Ireland has opted to form the βCommon Travel Areaβ with the UK which is a collective Immigration zone that is distinct from the Schengen Zone.