No, it's a straightforward service on one of the world's better low cost carriers. You will need ID to board but that's to check you haven't sold on your ticket, other airlines don't require ID. Arriving into EDI you are a domestic passenger for all airports in Ireland including the Republic. easyJet aren't dogmatic about sizes if you are very slightly out but you best conform to the sizes on the website. They are dogmatic on the number of bags, so you will need (e.g.) to pay extra for a front section seat for that second bag - there are various ways to do it. If you are within the hand luggage size (56 x 45 x 25 cm), easyJet will allow you to check it in for GBP 6, to save effort in security.

The land and sea route can be gorgeous in good weather, there is an even more interesting ferry service during summer via Campbeltown and Ardrossan, but easyJet is going to cheaper and massively faster. There are other providers from Northern Ireland to Scotland, such as LDY-EDI on Ryanair, Flybe Virgin Atlantic to Belfast City BHD (note easyJet goes from Aldergrove, Belfast International, BFS some distance from Belfast itself). Glasgow also has services from all 4 Ulster airports too, plus CFN, which puts in a claim for being the most beautiful airport in the world. Same carriers plus Loganair. In all cases the flight time will be very short, indeed shorter than the advertised timings.

Feel free to follow up, there are quite a few options here.