Not only did Alexandra not ask for help but she didn't want it. She felt she knew what was good and right for everybody and those who disagreed were banished from her sight. Again with Marie Antoinette she honestly just seemed to be ignorant, a luxury Alexandra did not have. I also believe that Louis and Marie suffered more for the sins of their predecessors than Nicholas and Alexandra, they could have done "some" things to prevent their fate, but instead they continued on the path they thought was best.
As for a public trial, Louis and Marie were found guilty and executed accordingly. Nicholas and Alexandra never had the luxury, and I feel if they had been allowed that then perhaps the world wouldn't have gone so Romanov crazy. Also if there had been a public trial more likely than not, the daughters at least would have been allowed to live. But instead they were secretly and brutally murdered, burned, had acid pored on them, and dumped in an unmarked grave in the forest.