As a woman travelling on her own. I am looking for experiences during your trips to Japan and whether there are cultural things I should know about to make my journey more pleasant. I am looking for a time next year when the weather would be pleasant as well. Thanks.
Certainly you should try onsen (at a traditional ryokan) as that is something special for Japan. Other unique things you might like to check are Kabuki, sumo and the food (including sashimi - good stuff, not "fast food" versions).
Travelling around as a solo woman should not present too many problems, the issues you will encounter are more likely to be as a result of difficulty with language or unfamiliarity with how things work. That is, off-the-beaten path can get trickier but the experiences can be very rewarding.