1) Targaryens are not immune to fire; it is known. GRRM has stated that Dany's fun on the funeral pyre was a ONE TIME exception to the fact that people burn.
To explain the situation with Viserys, I believe that Dany's comment was a shell-shocked realization that her brother was in fact, only human. Actual Dragons, like her 3 babies, are immune to fire, but Viserys in his madness believed himself (and taught Dany likewise) that he was an actual dragon. He's not the only Targaryen to believe something to this effect. Aerion Brightflame was a Targaryen Prince who believed that drinking wildfire would physically transform him into a dragon. He had similar results to any other human who comes in contact with wildfire.
On a non-believing oneself to be an actual dragon there are plenty of examples of Targyens being burnt. Summerhall is the big one here. A large portion of the Targaryen family died when that particular palace caught fire. Dany herself has also been burnt. In A Dance With Dragons, she receives multiple burns from various sources during her time as a feral dragon rider in training. She gets sunburnt and burns her hands and feet on the hot stones of "dragonstone" that Drogon makes his layer.
2) Warg are something we know little and less of.
Most of our information about how warging works comes from Varamyr Six-skins. We know what ethics wargs are supposed to hold themselves to (Varamyr fails on this front) and we know that its pretty bad for a warg to be in the skin of something when it dies. And we know that a Warg can "Live on" in a former familiar. As for the reproduction of Wargs, we only know that Varamyr has sired many children but none of them are wargs.
We also don't know how Valyrians controlled their dragons. Dany just knows that they used spells of one kind or another. It could very well have been something almost identical to warging (if not warging itself).
In fact the Targaryens have produced at least 1 warg/greenseer for certain in the form of Bloodraven/3-eyed crow.
Likewise the presence of Balerion the Cat may indicate that Rhaegar's daughter, Rhaenys had the gift of skin-changing and has been living on in her kitten, now a mean old Tom cat that knows way more than any cat should. It led Arya to the tunnel out of the castle and has been sabotaging Lannisters for years.
I've also been mulling over a theory that many of the Noble houses that use an animal sigil had a warg ancester with the creature in question as a familiar, hence the tendency of Nobles to refer to themselves as actually being that creature. Starks are Direwolves, Lannisters are Lions, Tullys are Fish, Arryns are Falcons, Targaryens are Dragons and Cleganes are Hounds.