This is a very nice site and I'm enjoying reading all the posts!
In my opinion, escape of the IF was possible on two occasions - once in Tsarkoe Selo and once in Tobolsk, and even then not all of the family could have escaped.
When they were in Tsarkoe Selo, as the govt. was falling apart, the empress had a window of opportunity to leave with the children or send the children away. She was cautioned repeatedly to leave, but chose to stay. Yes, the children were very ill, but as she was told, "the first to be carried out of a house on fire are the invalids."
The second opportunity, I believe, came in Tobolsk, when they were notified that the Emperor would be taken away, and she insisted that she go with him. Up until that time, I don't believe that the decision had been made to shoot anyone but the Emperor. If he had gone without her, she probably would have been arrested and imprisoned, but the children might have had a chance to be at the very least sent to the Crimea to their grandmother.
Once the Emperor and the Empress (with GD Marie et.al) left for Ekaterinburg the fate of the entire family was sealed.