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.
I agree that the first photos are liverwort. The little cups are the fruiting part that holds the spores. If it gets too rampant it can completely cover the soil and make watering difficult so I get rid f it too. Pull out or spray with vinegar.
The germinating seed looks like it could be some sort of elm. Seedlings in pots are common if any mature trees are nearby. You could dig that out and pot up for future bonsai. No need to get all the root. Germinating seedlings will readily grow even if the main root is cut.
Spring is coming and time for a repot for this one.