I think the alternate Northern line via airport is interesting, but you know politicians would interfere and insist stops be inserted to serve their communities, and we'd end up with two routes that fast trains get held up behind slow stopping services on.
I still think 4 tracking the existing Connolly-Malahide route would be to most cost effective solution. I think there are parts that would be tricky, but most are achievable. The sections in cuttings should be possible to widen by replacing the gently sloping cutting sides with concrete walls. Unfortunately, during the DART upgrade, no consideration of future widening was given, and I think all of the north side stations would need to be rebuilt. There were also a number of developments allowed close to the railway during the Celtic Tiger years, and some of those may have to be demolished to allow it. I still think this would be cheaper than a new route with a lot of tunnelling.
While there is no competition from air travel on these routes, the motorways is the main competition. It's not really cheaper or faster to go by train, so there's not a lot of incentive to take the train. The only advantage the train has at the moment is comfort, and at peak times between Connolly and Drogheda, that's very questionable.
If there was a true high speed service, and if the cost of making the journey by car reflected the impact on the environment, the train would become a lot more attractive, both in time and expense.
Operationally, I think two trains per hour to Belfast would immediately be viable, one non-stop, perhaps taking about an hour, and a stopping service retaining the current 4 intermediate stops, maybe taking 80 minuts. This could perhaps increase to 3-4 trains per hour in the future.
I don't know the Cork route as well, but it seems to be well subscribed with the currently hourly service, so I think a high-speed route could sustain higher frequency service. The high speed line could also allow improved journey times on other routes sharing the high speed parts of the line.