So what you have there is a very established JBP that needs a repot. The pot itself isn't too small, rather the tree is potted low in the pot - almost as if the tree has sunk 1" over the years since it was last repotted.
Answer to Q1: those are fruiting bodies from fungus. Don't panic - many fungi are beneficial. It just demonstrates that your soil is very established and has not been changed in a while...
Answers to Q2: if you keep bonsai in a general nursery, you get all kinds of random stuff popping up in your bonsai soil. Your first pic, I don't know what it is, but I get it too... it is a common groundcover plant. The leaves and stem feel fleshy - almost like a succulent. It in not particularly invasive, but I pull it when I see it. Photo #2 is a germinating tree seed. Not sure what type... but if I had to guess looks like an elm.