do European families rent rooms per week, month or season?

First, how do you plan to deal with the Schengen zone visa of only 90 days out of 180 days limitation?

Europe is a varied place, many countries, many cities. I'm sure you could find hosts, or something like Air BnB where you could stay in peoples' homes. But you will have a problem with the Schengen regulations unless you plan very carefully.

Step One, get visa
Step Two, pack cell phone. If you have TMobile or AT&T it makes no sense to spend money on a throw away phone in Europe.
Step Three, ask youself; "why do I want to live in someone else's home"
Step Four, start looking for cheap tourist apartments or hostels in the countries you want to visit.
Step Five, repost in GENERAL EUROPE

WOW! I asked a simple honest question and got nothing but rude, pretentious, snide remarks. What a waste of my time. I'll take your mockery and indulge in my own research and plan a balanced itinerary. Furthermore, I am quite aware of the Schengen area visa requirements.
(it's Dutch level 1,2,3 from the Nederlands Rosetta Stone)

I don't think anyone meant to be snide. I think the Dutch thing threw people off because you didn't mention the Netherlands as a place you plan to go on your trip.

Hopefully someone will come up with something to help with the "host family" question. Did you mean a host family that would give you lodging for free?

I did a similar trek back in 1977-78 long before Schengen. I traveled for 4 months and hit every country you listed except Hungary. I was 31 at the time. I stayed in hostels, B&Bs and pensions. The latter 2 were made possible because I met people in the hostels to share lodgings with in places where we were both going. I did meet a few people who let me stay at their places for free for a few days, one in London, one in Amsterdam, one in Florence and one in Copenhagen. But everywhere else was on my own dime, shared or not.

One of the great advantages of staying in hostels on the trip you described is that you can meet others who are traveling the same way you are. Back when I did that, I was really old to be staying in a hostel, but nowadays people of all ages stay in them.

I do think you will get more and better responses if you move this topic to the General Europe category. That's designed for queries that cover several countries. The people who monitor "all topics" will still see it and others who are planning to travel in a similar fashion to you will be more likely to see it in General Europe than anywhere else.

If you decide to dip into and out of the Schengen countries in your six months, a Schengen calculator might be useful in your planning. I've never actually used one, so I can't say one is better than another, but I found these online that you might try:

1. http://adambard.github.io/schengencalc/
2. http://ec.europa.eu/dgs/home-affairs/what-we-do/policies/borders-and-visas/border-crossing/schengencalculatoren.html

It does seem like many of the Schengen calculators are mobile phone apps. You can probably find many other options using your laptop both before you go and while you are there.

Good luck with your planning. If I weren't a bazillion years old and didn't have a dog I hate to leave in the kennel for longer than 2 months, I'd do something similar to what you plan to do. The only difference is that I pack for a week and a day, travel very light and do my own or have laundry done for me at one of the many self-service or "leave it and pick it up later" laundries. I couldn't carry a month's worth of clothes on my old back.

Sorry, but I don't see anyone being snide or pretentious. On the contrary, since you spoke of it being your first trip, we were concerned that you might not know about the Shengen rules. Many people who haven't traveled - or haven't in awhile - don't know. I guess the other poster assumed there was an autocorrect in the language you are studying because you aren't going anywhere where they speak Dutch - again, not meanly intended. To answer your question, in some Air BnB situations you do stay in peoples' homes, so that could be a resource to check out. I'm not sure what else we can tell you.

Wasn't my intent to be snide. Brief, but not snide. So more drawn out:

You might look into visas

You might look into TMobile or ATT vs a sim card in Europe. I think it might be cheaper and more convenient.

The concept of living in someones home is a little foreign to me. Have you looked at cheap apartments and Hostels?

If you post in GENERAL EUROPE you might get better responses.

I don't know about for a week or two, but you might check into 'couchsurfing'

See several types of accommodations discussed at https://www.ricksteves.com/travel-tips/sleeping-eating. While youth hostels might limit your stay to 5 nights or so, and couchsurfing hosts might also, other paid accommodations don't.

In my experience of traveling for six or more months (before the Schengen visa limitations), I rarely stayed anywhere for a whole week, except London, Paris, Vienna. Moving on more frequently can help you to see more of each country, a mix of large and small towns, etc. Day-tripping from a home base can also work, but at a certain point the return trip becomes a bigger waste of time than picking up and moving.

Also, I would describe a better packing plan as "two weeks worth of clothes" rather than one month.