Do different fighters yield different items?

I believe that, as you complete Classic and All-Star mode with more characters, more trophies are unlocked for availability. I have no proof of this though, only gut experience.

It is slightly more likely to get custom moves and equipment that match the character you're playing as.

I noticed that I tend to get character specific equipment doing All-Star mode.

As far as the trophies, the best way to get new trophies is Trophy Rush. Any trophy can drop there. I know that the only way to get all of the Final Smash trophies is to complete All-Star Mode with every character.

I have not found any evidence that only specific trophies drop during specific modes, though.

In Classic and All-Star, the custom moves and equipment you get are random, but are more likely to be moves or equipment for the character that you're playing as. In Classic, there is also an elevated chance that the item will be for one of the opponent characters.

Random trophies are have different rarities. Which mode you're playing doesn't seem to affect which trophies you get (aside from the non-random ones for completing achievements or completing Classic or All-Star as each character), although the difficulty does. There are six rarities of randomly acquired trophies, and each of those rarities corresponds to a base price in the Trophy Shop. The most common trophies cost 300 coins and the rarest cost 2500 coins. Playing on higher difficulties increases your chance of receiving rarer trophies, although the exact algorithm is unknown.