1- They probably are snails eggs. Like mentioned above, little tiny sack (clear) with little dots. Probably came in on your plants, or you might even have some in your tank.
2- Tiger barbs are schooling fish. You should have at least 6 together. The develop a pecking order. If there are too few, they will pick on the other fish. I could almost guarantee that is why the TB is messing with the cory.
3- Cory's are schooling as well. I personally have no more details on them though. I am sure someone else will come along.
~30 Rummy-nose Tetras, ~30 Balloon Lamp-eye Tetras, 2 Bristlenose Plecos, 4 German Blue Rams, 1 Rainbow Shark ~15 Amano Shrimp, 8 Roseline Sharks, 3 Clown Loaches, 2 SAEs