Having just returned from our trip, I will try to provide a description of how we planned it and how things worked out.
When we arrived in Vienna we purchased a topographic map of the Gesause region at a book store. With this in hand, we decided to take a train from Vienna to Gaishorn am See, to the south of the park. The train takes three hours and isn't very expensive, especially if you buy tickets more than a week in advance.
In retrospect, I'm not sure Gaishorn was the best choice, because it doesn't have any supermarket, although nearby Trieben has two. But it did have a hotel and some restaurants, and a trail that leads to the Modlinger Hutte, the hut we planned to sleep at for our first night. (it was very pretty, and the staff were very nice)
The area of the park is one of valleys and mountain ranges, so that our plan was to hike downhill most mornings and uphill in the afternoons. In July the sun sets very late there, so there's plenty of time for a long rest at noontime.
We also ended up doing a lot of hitch-hiking, which is not for everyone, but people were very friendly and took us willingly. There are some bus routes on the roads the cross the park, but they're infrequent.
From Modlinger Hutte we descended to Johnsbach, where we had lunch (at a lovely hotel-restaurant, Gasthof zum Donner). We hitched a ride along the valley road to its eastern side and ascended to the Hess Hutte that afternoon, and slept there.
From the Hess Hutte we descended to the main park valley. Other travelers told us that this path - which involves ladders and safety cables along a cliff - is not recommended in this direction, but we thought it was OK. From the valley floor we hitch hiked to the park pavilion, had lunch, and ascended to the Ennstaler Hutte, where we stayed the night. This hut had an amazing view- one of the best I've seen, anywhere around the world, and they had a particularly cheerful staff. It's the "don't miss" place of the park, in my opinion.
The next day we descended to the park pavilion. We planned to ascend to the Haindlkarhütte, but were too tired, so we hitch hiked back to Johnsbach where we spent the night.
The next morning we ascended back to the Modlinger Hutte, and though we were planning on going straight to Gaishorn am See (where we left things and had return train tickets) we instead walked to the Oberst-Klinke-Hütte and hitch hiked from there down to the valley floor, and from there walked to Trieben.
1) Sleeping arrangements
The huts were crowded, but not ridiculously so. We signed up as (foreign) members in the alpenverein - the Austrian Alpine society - which provides a variety of benefits, including discounts at the huts and, supposedly, precedence in sleeping arrangements, but we didn't need the precedence. We didn't manage to catch any huts by phone in order to reserve our place, but it wasn't necessary.
The Austrian Alpine Society membership discounts were honored everywhere, although the staff of the Hess Hutte may have overcharged us under some very dubious circumstances - but since there's a language barrier involved, take this with a grain of salt.
Bring either a light sleeping bag or a "blanket cover" for the huts. Not all the huts have showers. You can find out about them using the hut finder site which gerrit, which was also one of the most useful planning tools that we used early on.
Two useful links: The UK branch of the alpenverein, and the Vienna English speakers branch.
Not knowing German was a liability both in planning and execution, but everything turned out fine, nonetheless. I have never been to huts with so few English speakers, though- it will require more boldness to enter into conversation with other guests, since all the conversations around you will be in German initially- but it can be done, if you want.
3) Equipment It can rain in the park, even in July, quite suddenly. We were hailed on. Bring rain gear. Walking sticks are also essential for a park which has almost no level walking.
I'll add more information if anyone has any questions- we had a wonderful time and recommend the area. And we'll probably try another hut trek in Austria in the future!