Did Rand really threatened Juilin Sandar in the Shadow Rising ? I've always thought it was just Lan who did it and mentioned Rand name to get an effect. 

Seanchan new the Dragon was reborn, and they also have the prophecy that the Dragon would bow before the Empress, may she live forever.

No one knew how Turak died.

Hawkwing fighting for Randland is a good question. Tho it isn’t clear how many people saw Hawkwing.

The threat to Sandar may have been form of speech, not a real threat. 

In most country's that Rand occupied he was quite good with politics, true there was rebellion in Tear and Cairhien but they were mangable. When he occupies Andor he manage to turn majority of the nation against him. How did Rand manage to to turn so many against him ? What did he do wrong ? 

There were rumors being spread that he'd killed the good queen Morgase.

He also publicly declared amnesty for male channelers.  In a world taught to fear men that can channel, this would be received by the population like someone had just placed hundreds of poisonous snakes inside their homes.