No. You can and should harvest every day. The formula for when they become available for harvest is a bit weird, but it is based on the time you go to bed. The later you go to bed, the sooner they will be available the next day. If you go to bed at 2am they will be available at 1240 pm the next day (sorry, iridium quality, I had that off by 10 minutes), and if you go to bed at or before 6pm they will not be ready until the second day. For that reason I suggest harvesting mushrooms be one of the last things you do in any day so that you can harvest daily.
Additionally, yes, the only quality you can get are regular quality.
There are two main benefits to the mushroom cave. First, the fall forage, dye and field research bundles require mushrooms and the exotic forage bundle can be filled with mushrooms and the cave makes those easy without having access to the Secret Forest. (Many argue that the fruit bat cave making it easier for the artisan bundle is better than the mushroom cave for these bundles.) Second, it is good for making life elixirs. Having a stack of those is nice because they heal you to full no matter what it says (it says it only restores 200 energy and 90 health, but this is incorrect and I don't know why there is this discrepancy). I usually just store all my mushrooms for a while, then make a batch of life elixirs, and when I have a nice stack (10 or more) I bring them along when I go into the Skull Cavern for when I need healing and don't want to use food that gives buffs. (I typically bring along 1 stack of food for buffs, and 1 stack of food without buffs for healing.)
You will end up with a lot of extra common and red mushrooms, though. Once you get mushroom trees, you just may end up with a lot of extra purple mushrooms too. When I have a mushroom cave I sell all higher quality common, red and purple mushrooms. I also will occasionally sell any excess I have of the regular quality ones too, if there are a lot more of those than there are chanterelles and morels. It's money, but it's not a huge money maker. If you regularly hit up the Secret Woods in Spring (for morels) and Fall (for chanterelles) you can cut down on the excess you'll have of red and purple mushrooms by making more life elixirs.