For the mountains part of your question, how experienced are you in the mountains? Fitness? Technical or nontechnical only? Altitude? Time of year? Are you weather/avalanche savvy? Experienced with crevasses? Language really doesn't matter much up there ;-) There are lots of huts and they will all meet some minimum standard.
A couple of possibilities I have first hand experience with:
Ankogel from Mallnitz. You take a cable car up to Hannoverhaus, stay overnight then early in the morning up to Ankogel (3200+). It is possible to get up Ankogel, have lunch at the summit, then descend and get the train to Vienna all in a decent day. (Assuming a reasonable degree of fitness.) But you do have to climb--not with a rope, but it's hands and feet in some short sections with considerable exposure. Watch the weather closely here.
Moserbodensee. This is up from Kaprun in the Glockner Group. The usual way is to get up to Kesselfall, then buy a ticket for the bus and a big outdoor lift up to Mooserbodensee--there is a water power dam here and a visitors center. It's sort of family oriented, but few people venture across the dam and into the mountains. From the lake/dam you leave the tourists and hike up to Schwaiger Haus. It's not far but it is steep and the altitude is a factor for some people. Non technical. From there a very early start in summer can take you to the top of Wiesbachhorn (3500+), but you probably need to be roped up and use crampons. From the summit you can see Glockner and the alpine road. Even without the summit this is spectacular, esp sunrise over the valley from the hut. I'm not sure how this one would work without a car. You could probably get a taxi from Kaprun without much grief.
In the Oetztal, there is the Grosse Venagtspitz. There is a bus up to Vent from the rail station, then cable car up the lower part of the valley. Then hike across to Breslauer Hutte, following a very good traverse path over several glacier run off rivers. From the hut a very early start (~4.00 am) can get you up to the summit--but it's a serious glacier hike (ropes, crampons, crevasses, altitude). Even not going up, though, you can hike up the ridgeline of a moraine to the toe of the glacier and enjoy the 360 views of glaciated valleys and peaks. From there, you can hike down a slightly different route where two valley come together, through a Halflinger horse breeding farm, and back to Vent. Bus to the train station. Without the peaks and using the cable car to get up the steep and boring first bit, and wearing decent boots, this is a completely nontechnical two-day tour with spectacular scenery. Vent itself is a charming little resort town with all the comforts.
Summiting Glockner is not a trivial project. It's not a 'hike', but a mountaineering expedition. I've not done it myself. But I would not go up there unless I were going with a very experienced and sensible group.
In any case, for the high mountains, get current info from the hut master on weather, crevasses, etc. You will need hut reservations as well. And be sure you have rescue insurance. If anything goes wrong with you or the weather and you have to be rescued, you WILL be charged. A lot.
This might be a resource for you in planning:
http://www.alpineclubvienna.org/archivedevents.php
Lots of trips with notes on access, times, huts, etc. for many different trips.