The two trails you mention (Frankenweg and Jakobsweg) are not mountain trails, and instead of shelters or mountain huts, you will encounter regular hotels and bed&breakfast (Pension in German), and plenty of them. Germany is a densely populated country - you'll probably come across a village at least every 2 hours, though there may be some rare stages of the trails that cross sparsely populated areas; you should plan ahead a little.
These accomodations are usually open the whole year. Expect to pay around 25 EUR for a single room in a Pension, and at least 35 EUR in a hotel.
You'll typically only encounter shelters and mountain huts if you plan to go hiking in the Alps, and many of these are closed during the winter, though there's usually some sort of emergency access (and some shelters have mainly this function). They usually have only dormitories, and reservations are very much recommended; they'll never turn you away outright, but in the worst case you'll have to sleep on the floor somewhere.