There is no minimum time between the two stays. However, the overriding requirement is that your visit is for tourism purposes. An immigration officer has a discretion to refuse admission if he/she is not satisfied that the purpose of your visit to Ireland is genuinely for tourist purposes. While there is no minimum time in law between two tourist visits to Ireland, two visits in close succession for the maximum allowed time of 90 days may lead an immigration officer to a suspicion that the purpose of the trip is not genuinely for tourist purposes. The immigration officer would therefore be more likely to ask you on your second arrival in Ireland what activities you intended to undertake in Ireland. The immigration officer might ask what you did on your first trip to Ireland and why you want to do either the same activities again so soon, or else why you did not get to 'see the sights' on the first 90 days in Ireland that you now want to do on the second visit. However, there is no reason why you could not satisfy the immigration officer that your second visit to Ireland was also for tourism purposes. It is not unreasonable to spend 6 months in a country as a tourist, but it is also not unreasonable for the immigration officer to ask some questions about it to satisfy himself/herself that the second visit is also for tourism purposes. In this regard of course, hotel bookings, tour bookings, planned itineraries for your first and second visit of Ireland, would be recommended, as these would show that the purposes of both of your trips to Ireland were for tourism purposes. If you do this, you will not have any difficulty entering Ireland for a second tourism visit of 90 days.

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It depends on what exactly you are doing. From what you have described, it is unlikely to be a breach of tourism-purpose visit - if you are not interacting with Irish companies or people in a work related manner or engaging in any trade or commercial activity which is taking place in Ireland, then what you do on your laptop would not be a breach of your tourism visit in Ireland. When 'working remotely' for your American employer, the work itself would be deemed to be taking place in America. If however, whatever you are doing on your laptop requires you to be physically present in Ireland to do the work remotely for the American employer, then this would be an indication that you may be considered to be working in Ireland for your American employer, in which case you would require a work visa as the American company would be conducting business in Ireland. That this not seem to apply to the circumstances which you have described.