In this case, "badly" is the correct usage; as an adverb, it describes how you miss someone. Just as a note, "badly" here means to an extreme degree, not like the opposite of good.

In most cases referring to relationships, "fight" means an argument; using only words. However, the word "fight" can be completely dependent on context; if your second sentence followed a more serious idea like "Jake pushed his friend, and they ended up having a fight," then it would be more likely that it involved an actual physical confrontation.

That URL doesn't seem to work for me; are you sure it's copied correctly?

He says "Three and Oh to <player name>"; I can't even understand the name he's saying, but I don't follow baseball at all. It's also possible he's saying some sort of nickname. In this case, "Oh" is slang for "zero"; the pitcher has thrown three balls and zero strikes.

In this case, "into" means he has a significant interest in it. It can be used to show interest in any topic, really: "She is into soccer," "He is into computers."

In this context, "body" usually means "thickness" or "fullness." Since the speaker says "or more salt?" as an opposing choice, it is likely that they are referring to how thick and flavorful the soup is.

"Find" means "to locate" when referring to a physical object. So, in terms of the wallet question, "Did you find it" does indeed mean "Were you able to locate it?" To say "Did you make an effort to locate your wallet" you could ask "Did you look for your wallet?" In that case, "look for" specifically talks about the effort. "Find" works similarly when talking about jobs; to tell someone "Go find a job" means to get employment somewhere. It is similar to the phrase "look for a job;" both mean that you are trying to gain employment. Hopefully that helps; please let me know if you need any more clarification.