A couple of times I've seen two slimes in the mines breed. One of the slimes makes a heart bubble and another makes an exclamation mark. I don't really pay attention so I don't know which is male or female. And afterwards a little slime pop's out. I wanted to find more information about this but I couldn't find any.
Slimes will certainly breed in the mines! (Where else do baby slimes come from, silly.) The trouble is that aside from the top floor where the entrance/minecart/dwarf are, all levels of the mines are non-persistent and change contents and monster spawns between visits, so none of the little slimelets will be seen again once you leave that level. But to paraphrase Gertrude Stein (And I'm sure she would have written on slimes had she but known!), a slime is a slime is a slime, and they act the same no matter where they appear.
We have a whole thread dedicated to people who breed "wild" slimes, both with and without slime hutches. http://community.playstarbound.com/threads/the-magical-thread-of-slimy-awesomeness.127962/ Possibly anything anyone would ever want to know about slimes can be found there.
They will behave the same with or without a slime hutch-- the difference is that the slime hutch makes the population in it persistent, which means that the babies will remain after you leave/return. They'll breed in the Mines, the Skull Caverns, and the Secret Woods, anyplace Slimes occur naturally at all, but since those populations are randomly spawned each time, they don't stay there. Also in general (with very rare random exceptions) those naturally occurring Slime populations will all be of the same sort based on location/Mine level, so you won't get any interesting cross-breeds for those brief periods that the baby Slimes exist. There is /one/ exception to this rule that I know of-- a very rare glitch where when two Slimes breed and the baby is spawned at the exact second you leave that screen, it can appear in the new location. I've had baby Slimes 'follow' me out of the Secret Woods and into Cindersap with that glitch, and once they're in that location, they become permanent residents the same as if you were breeding them there! If you were very lucky and could time it properly, you could theoretically (Can I use that term regarding Slime breeding? I can. I have.) force a breeding population to spawn one baby at a time by using the glitch, no incubators or Slime Eggs needed.
I've seen them breed 'in the wild'. It doesn't happen often, because you have to have both a male and female slime of the same type in close proximity to each other and not actively attacking the player but close enough that you can see them. It is more likely to occur on floors you have to clear to generate a ladder which have lots of the same type of slime.
I posted this in the thread of randomness. It wasn't the first time or the first playthrough in which I witnessed this and I never built a slime hutch in over 300 hours of playtime. That's because I never killed enough slimes to get the ring. Every farm was abandoned in year 3 because I achieved everything I wanted.
Neat! I'd only ever seen it happen in the wild once I'd had the hutch built and the subsequent cutscene you get from it. I'd assumed that was what triggered the behaviour. Good to know.
As far as I've been able to tell it's random; there's no slime mating season. Since they have to be within sight of one another to fall in love, you can help encourage it by keeping them close together (confined to a small space or lead them near each other). Attacking you seems to take priority to les affaires de cœur though, so if you're being a distraction they don't seem as inclined to do the slimy thing.
I think I have to actually disagree with this part to a degree. I have slime in my farm cave and when I walk in and they all come at me, I've had a couple of them just stop and go get... er... slimier... ummm.. yeah... haha But, they are more inclined to go for you than each other, but it is possible.
I've spent a few seasons trying to encourage a struggling colony of them up at the Railroad, and I can lead them towards one another by getting them to chase me, but they don't seem inclined to pursue other interests if I'm still close enough to target. That may just be something subjective. I always did want to be the center of attention!