I have been in a version of your boat, albeit a long time ago, and I am sorry you are struggling. Language schools with homestays are very different from the typical study-abroad experience in that there is less of a built-in community and not so much structure around out of school activities. And let’s face it, struggling to communicate all day is exhausting in and of itself. I echo a lot of the advice here – especially letting your parents in on what’s going on and making sure you are eating enough.
A couple of things that MAY help… Would you consider teaching English or being a conversation partner for a few hours a week? The teachers at my school were there to teach (and I think they were told as much!), but the people I met through this avenue actually were excited to hang out with me and to share their culture. Are you interested in learning calligraphy, painting, flower-arranging, cooking, or anything like that and could you find a class? It’s much more relaxing than drilling in a language and again, people are often happy for your interest in their culture. Is there somewhere you can exercise? You’ll feel better for doing that and you may also meet some people that way.
Try to talk to your family. Perhaps your Japanese “mom” can teach you to cook something and you can make a meal for the family. (Think tacos – something simple and participative – and it may give you a chance to show them how a meal can be “customized” for the different diners with different tastes.) Maybe they have favorite TV shows or something that is not so stressful to share. While it sounds like the living arrangement is hard for you, I am also imagining that it feels awkward to them too. I used to play with my family’s young kids, and it was silly and fun for all of us but most of all, it created enough of a connection that when I came home, they were excited to see me, and boy, did that help!
And yes, those day long and weekend excursions can really break the monotony, so make those plans.
Lastly, one of my favorite ways to deal with kind of situation is to write about your experiences, especially if you can find the humor in what’s making you nuts, and share those (FB maybe?). It might allow you to experience them more positively and it will allow you to connect with friends over your adventure. So that when you have a plate of food that seems repulsive set in front of you, rather than feeling like you’ll collapse into a heap of tears, you take a photo and mentally start composing the hilarious post you’re going to write about eating in a land where nothing is off-limits for the dinner plate. (In the pre-internet days, I developed a “menu” in my journal while traveling once that included items like “tough as leather chicken”, complete with a description of how it would be prepared – and every item on the menu was prepared with the same spices.) You can post about a trip on the subway or shopping at a local market. Your friends will get a vicarious thrill and you’ll be connecting with them. It is possible to do a mind shift.
Mostly though, hang in there. I know it’s hard and that when it’s hard, a small effort feels like a big one. But you’ve got this!