Actually, that looks pretty doable. I'm a cyclist and though I've never driven a car in Eire, I've been to all those places and overnight spots several times. The only changes I'd make are:
1) On day seven, see The Burren before going to the Cliffs of Moher and Doolin to save on backtracking. The Burren is northeast of Doolin, whereas the Cliffs are south of and quite nearby Doolin.
2) Perhaps switch days eight & nine. The Aran Island ferry leaves from Doolin, so no need to backtrack from Galway. Leave your car in Doolin and make a day trip visit to one or two of the islands. Alternativey, you could drive west from Galway to Rossaveel and hop the ferry to Inishmore, the largest of the Aran Islands. Check the schedule though to verify that you can catch the ferry back the same day. You may have to overnight on Inishmore and take it the next morning.
The link to the Aran Island Ferries is: [http://www.aranislandferries.com]
The Doolin Ferries link is: http://[www.doolinferries.com/]
3) Once you've returned to Doolin, then drive to Galway, which is in Connemara, (probably an hour or so away) for an overnight. Then, drive the N59 loop through the wonderfully scenic Connemara region, returning to Galway and stay in either Galway or Ennis. If you want to cut the N59 loop a little short and see even more dramatic scenery, then head through the incredibly tranquil and pristine ten-mile stretch of the Inagh Valley on the R344, hooking up with the southern or northern portion of N59 west of Galway City.
Lots of B&B, trad (pub music) and hotel and dining options in both Ennis and Galway. From Ennis, I'd guess it's a two-hour drive to Shannon.