I think you've basically got it, but I assume you won't know which the HNO3 or AgNO3 are initially, meaning you'll have to react some compounds first to find out which they are. For example the two unknowns that form a white preciptate have to be AgNO3 and NaCl as mentioned, but you won't know which is which until you react each of these with another unknown to form the yellow precipitate (KI as mentioned) and the NaCl will form a solution with KI, but no precipitate. Now you know those three, you know the remaining two are Na2CO3 and HNO3, so you can react each of the two unknowns with AgNO3 - the Na2CO3 will form a yellow precipitate of AgCO3 (rememeber most carbonates are insoluble, so the colour doesn't actually matter here) - there may be another way the spec wants you to do this however.
You are correct with what you said, but I think there is a fundamental logic error, because you can't just add HNO3 or AgNO3 if they are all unknown initially.
And I haven't come across any tests for HNO3 or AgNO3 explicitly, but there may be some.