If it was addressed in the show I didn't catch it, but doesn't having two brothers, one a firebender & the other an earthbender imply that their parents and/or grandparents are from different nations?
I thought that was a nice subtle way to illustrate just how much has changed in 70 years and indeed, what Republic City is all about.
That's right -- according to what I've read, they're the children of a mixed marriage. And you're right, it does illustrate the melting-pot nature of the Republic.
Something else that caught my attention; if matey-bloke-who's-name-I've-forgotten is currently the only the only airbending master, does that imply that his two siblings that Katara mentions are either waterbenders, less accomplished airbenders or not benders at all? Given their parentage the last option seems the least likely.
Tenzin's older sister Kya (named for Katara's mother) is a waterbender, and the middle child Bumi is a non-bender. At this point, the only living airbenders are Tenzin and his three children Jinora, Ikki, and Meelo -- and of course Korra, potentially (and maybe the fourth child that Pema is carrying, if Katara's intuition is valid).