I agree; Emma dislikes Jane because she is as beautiful and intelligent as Emma, and has made better use of her talents--reading, playing the piano really well, etc., so she's an unspoken reproach to Emma for her laziness. Also, Jane is a rather quiet, reserved sort of woman, probably not really Emma's type. So, jealousy, reaction against other people's expectations, and personalities that don't mesh particularly well.
Emma hasn't met Mr. Dixon or his wife, Jane's friend. She's imagining Dixon's attraction to Jane, mostly because Jane didn't go to Ireland with the Campbells to visit the newlyweds, maybe reinforced by something or other the talkative Miss Bates has said.
The 30,000 pounds would be Emma's marriage settlement. How these would be used varied. Married women couldn't own property, so if no other arrangement had been made, the money would become the husband's, and the expectation would be that he would use it for the joint good of the family. The prudent course would be to live off the income the money produced, not touching the principal. It was possible, and prudent, to tie up a woman's money so the husband couldn't touch it. The woman had a "life interest" in it, and could enjoy the income, but the principal would go to her heirs.
I don't remember if you read the two Wilkie Collins books with us. In The Woman in White, the villain marries the heroine for her money, and spends it all. In The Moonstone, someone wants to marry the heroine for her money, but doesn't because it's tied up, and he needs more than just the income to get out of his debts.