What if, wounded by the Morgul blade, Frodo had fade before reaching Rivendell?

[dancing spawn of ungoliant]

Take it to Sauron, even though he hasn’t received the order to? Wait for the Ring-wraiths to find him?

Since Frodo wasn't travelling alone, I think we don't have to worry about those options. I assume that his friends wouldn't have let Frodo do anything that would have hurted him or possibly the whole M-E.

In a more comical way, could we see it float in the air where Frodo is standing

Wouldn't that be a sight!
Maybe once the shard of the blade would have reached Frodos heart and "done the job", it would have disappeared (well, like the whole story would otherwise follow the laws of physics ).

Would the real Frodo, the good one, be gone forever or would he be trapped in this evilness and only able to watch?

Like he had a helpless alter ego? I'm not sure if I believe that a piece of an evil sword can wholly change one's personality, but I don't think it could have made Frodo schizophrenic or something like that. It might be a bit like someone would have casted a spell on him (like in Theoden's case, perhaps?). If there had remained, after passing into the world of the wraiths, any good left in Frodo, it might have been something unconscious and it would probably have faded quite quickly.