For the climate in October,
take a look here. It's in German, but you can use Google to translate.
Generally speaking, a few refuges stay open until sometime in October, and some that are close to a road may be open all year round, but a lot of them will already be closed. A little further down there are more places to stay.
Hiking in the Texel Group is still possible around that time, and if you want to go for a long distance route, check out the
Meraner Höhenweg. Along most of the route, there will still be enough accommodation options available, varying from hotels to farm stays to mountain refuges.
Hiking the whole trail, which easily fits in 6 days or less, depending on what distance and elevation difference you're comfortable with, means that the crux will be the day that you go over the Eisjöchl, the highest point on the route. That may very well still be easy in October, but there can also be lots of fresh snow on the pass. A further small complication is that the Stettiner Hütte, next to the Eisjöchl, won't be open, so you have to plan your trip such that you cover the distance between Pfelders and Eishof Pfossental in one day.
To take this into account, you can plan to do the route starting in Pfelders, going counterclockwise. If there is too much snow or the weather is too poor to cross the Eisjöchl on the first day, you can still change plans: If poor weather is the only problem, then perhaps waiting a day is the solution. Or just go clockwise instead, and instead of hiking the whole circuit stop at Katharinaberg (if you left your car at Pfelders, you can return by public transport). Either way, if you change plans, you've got to make a lot of phone calls to change your reservations.
By the way, in Pfelders I recommend
Pension Rosmarie, and on the south side of the Meraner Höhenweg, I really like
Patleid.
If you go right at the start of the month, the Prags-Sennes-Fanes natural park might be another area to check out. At the east side of the park, a road goes up to Plätzwiese, around 2000 m or so, and there are a couple of accommodation options there. If you arrive early enough, you can use the afternoon to visit Dürrenstein (a steep hike, with 800 m elevation gain). The next day you can head west, into the park, to the Sennes Hütte for example, assuming that it's still open. After that, there are several possibilities.
Good luck,
Rob