Hi there,
For your 1st SW tank it isn't recommended that you start that small. The bigger the tank the more room for error you could say. It's possible to do, it just will require more work and constant monitoring on your part.
I think 20g High or 30g standard are perfect starter sizes if your limited on space.
But if you insist on a Nano I would say go for the 15g standard. I've been eyeing one on Orlando's Craigslist for the last couple days I think they have 3 and are asking for just $15.
$$ wise it won't cost barely anymore to start the 15 or 20g then it would the 10g.
Live Rock (LR) helps with biological filtration. Also some fishes need places to live or hide in the tank so having rocks helps with that too.
You don't HAVE to buy LR or Live Sand (LS), but it helps get your cycle going. Some people get Dry sand & Rocks and use chemicals and other products to cycle their tanks. I don't know details of the different methods of cycling since i got my tank already establish. I'm sure you can search the forum or maybe some1 else will chime in.
I have a 29g FOWLR and all I have is a HOB filter and it's been running for over 2 years.
I've been thinking about starting a 15g Nano so I've done a lot of research on cheap but effective equipment for it. So let me know if you would like any recommendations!