Spontaneous in Japan

I started to plan a trip to Japan that combines visiting cities and hiking in national parks, and I have a few beginners questions.
I have a rough idea of my route, and I don�t want to commit to a strict schedule in advance.
1. Can I look for vacancy after arrival in cities / towns / villages? ( Unfortunately I don�t speak Japanese, English only)
2. Is It OK to ask to extend my stay if I like an area and my Ryokan / hotel?
3. How can I travel in trains if I don�t know my next accommodation � can I take my baggage with me on trains, and not send it over to my next accommodation?

Thanks in advance for your help,

Ofer

This depends very much on when you are travelling. If it is a season when Japanese people are likely to be travelling (August, Golden Week, New Year, or the cherry blossom season or autumn leaves season in some areas as well as placed with a particular festival happening) then you may find that everything in that area has been booked out well in advance.

If it is an ordinary week day then you should be fine, weekends can be tricky sometimes. Also most hotels don't like you just turning up without at least a phone call in advance, even if it is from down the street.

Can I look for vacancy after arrival in cities / towns / villages?

Many tourist information desks can help you make same-day reservation. Of course, it comes with the obvious risk of not finding any vacancy in your budget range. Saturdays in tourist destinations are busiest.

Just showing up at a accommodation and ask for vacant rooms can be done in case of hotels, but it is not common in case of ryokan.

Is It OK to ask to extend my stay if I like an area and my Ryokan / hotel?

Yes, assuming that they have continued vacancies.

How can I travel in trains if I don�t know my next accommodation �Ecan I take my baggage with me on trains, and not send it over to my next accommodation?

Of course! Most tourists take their luggage onto trains.

My question was actually similar to this. I have some places I plan to go, but have no idea when.
What I plan to do is use a guest house in Osaka as my main home away from home. Then, when I travel somewhere else that is inconvenient to travel back to Osaka, I'd rent a hotel for a few nights to enjoy the scenery in that region.
I hope it works! It's my first time in Japan, so we'll find out, considering I'm on a tight budget. :)