Ag+ and NO3- is not just going to combine to give you solid silver nitrate because it is totally soluble in water, once any solid forms, it'll just dissolve and form the hydrated ions again.
H+ + Cl- gives no reaction because remember, if they did react, they'll form HCl, and you and I both know for a fact that HCl is a strong acid, so it is fully deprotonated in water, so if any forms, it'll just dissociate into Cl- and H+ ions again.