1. The meaning is the same but there is a regional difference. 'Show up' is a predominantly American usage: as I understand it, American English speakers use 'show up' meaning 'arrive' for people, and 'turn up' for inanimate objects (for example, if you've lost something, you might say "Don't worry. It'll turn up eventually"). British English speakers use 'turn up' in both contexts: a BrE speaker would say "He turned up late" whereas an AmE speaker would say "He showed up late". I'm not convinced by Denis' idea that 'turn up' implies that the arrival is unexpected - I think the meaning is the same. 2. 'Boil down to' is more informal and possibly suggests a greater 'reduction', but the meaning is more or less the same. 3. Interesting. I was on the point of telling you that you couldn't use 'turn in' in that context. However, I checked in a dictionary, and it turns out (another phrasal verb!) that you can. I had never come across this usage before, which appears to be North American. I would normally expect the transitive 'turn in' to mean 'relinquish', as in "The police ordered them to turn in their weapons". A synonym here would be 'hand over'. However, if an innocent member of the public happens to find a passport in the street, they might hand it in at the police station (not hand over). This is because there is less obligation and no suggestion of having to relinquish the object. So, teachers might tell students to hand over something that isn't allowed in class (a phone for example) but they would normally ask students to hand in their work. A speaker of AmE might be able to tell you whether it's common to also use 'turn in' in this context. I would avoid using it in a test situation, just in case the marker was unaware of this usage. NB Note that 'work' is uncountable in this context: The teacher asked them to hand in their WORK.

Thanks a lot!

to turn up has an unexpected connotation.... to show up means to be present... Overall though, these two idioms are pretty similiar. The same could be said for "boil down to" and "come down to"...the former is a little bit more imaginative and informal. Both "turn in" and "hand in" can be used interchangeably. All of them sound quite natural in spoken English.