Nobody wants to be told not to do all the things, but some times being selective can really heighten the experience! You don;t want to end up driving on bad roads for five days in the rain. .. .
I really can't think of a way to pull this off, Ireland is a pretty subtle country it takes time to absorb some of the quirkiness, plus I really think you should spend at least one afternoon/evening in a good pub, get chatting to people.
So first day is going to be Dublin it seems. Have you ever been to a Victorian Gaol? Check out Kilmainham if this sounds cool. (it is) Guiness is also nearby, which has a great visitor centre, but doesn't tell you much about Irish culture. I like Chester Beatty Library swell, the museum on Kildare st, and if you like contemporary art, the rha on ely place. Check out stephens green for some fresh air (theres a butlers chocolate coffee shop near the main entrance) and the rha is near the other side.
STAY CLEAR OF TEMPLE BAR AT THE NIGHT TIME AT ALL COST.
Check out south william st/exchequer st/fade st for nice restaurants bars.
Dublin- Newgrange is very accessible, can even be done as a day trip, via the boyne valley drive which is gorgeous Newgrange is as old as the pyramids so well worth seeing, trim castle is so near as well and is a great norman settlement. You could plough on to Galway from here or go back to Dublin for some r+r. FRINGE festival is usually on around Septmember in Dublin swell so there'll be plenty on in the evenings.
Galway-Aran Islands can also be done as a Day trip. This would also bring you to connemara, an Irish speaking area of Ireland, and then hopefully you'll be back in Galway in time to go out for a nice meal and a walk around, galway is also full of lovely pubs. Aran Islands are VERY SPARSE, so be ready. It's all about the rural charm.
Next day drive via cliffs of Moher to dingle peninsula, (not sure if you'll get out to see fungi at that time?), this will be quite a day of driving so bring a few snacks so you can stop and take in some scenery. You could also stop at the ailawee caves on the way if you had any interest? Big huge underground caves, you can just walk into them with a guide and they go on for miles.
Next Day start making a move back to Dublin, what time is your flight? You'll need to be at airport 3 hours before. You can drive via Cahir Castle and/or rock of cashel (which can actually be seen from Main road) if you have time, and this route is only 29 minutes longer than the main road.
Try air bnb for accommodation, seems to be taking off.
Finally above all, make sure and try loads of nice food, that is definitely one of the best things about Ireland, particularly the cheese. For a nice restaurant in Dublin serving Irish food try 'vintage kitchen' but you may need to book in advance if its for a weekend. Its right next door to mulligans, a real classic pub in Dublin.
Go forth and Good luck!!