Does visiting a random player have any benefit

Rare dragons are certainly less likely to be bred when they're an option. It's all a matter of chance, though there's been some unproven theories that the dragon you put on the left side of the breeding menu determines the order of elements of the egg. It doesn't seem to hold true.

Generally assume the breed will yield a random possible combination, and note that rare dragons are very rare, I'd say maybe less than a 10% chance. I don't think there's any formal numbers. From experience only Rare dragons have a different breeding chance, everything else, assume an even chance of all other possible results. If there is a difference, it's not large enough to fuss over.

Special Event dragons also tend to be "Rare" at approximately the same rate as the Sun/Moon/Rainbow dragons. Generally you'll want to stop everything and start trying to breed time-limited dragons immediately until you get one, or two if you want a breeding pair.

Note the Epic Breeding Island advertises that you have a higher chance of rares. It doesn't seem extremely significant; rare dragons are still rare on the breeding island. In fact I haven't noticed a perceptible difference, but if you're trying to breed for rares you might as well try the island first.

Also note that breeding two dragons always gives you that same dragon (best I can tell), so breeding two Rare Rainbow dragons has a 100% chance of giving a rainbow dragon.

You get coins for it, your friend doesn't. For some reason it's always somewhere between 500 and 7,500 coins per party hat. Your level seems to determine how many coins you get. As Ebongo says it appears to be roughly X² * 10 coins where X is your level, but not exactly.

The coins respawn daily and friends with more habitats have more chances to have coins, but the max seems to be around 5 party hats per friend. It's just a little bonus to encourage you to visit your friends daily.