Did Major Gilbert survive?
In the Light Novel the anime is based on,


   he did survive. He woke up before Violet and told everyone to keep it a secret from Violet so she could grow as a person beyond being a emotionally-stunted loli super soldier while he advanced upwards in the army as part of a spec ops team. He made his grand reappearance on the train to save the day.  


On the other hand, the anime


   left it vague because it's kinda a crap plot motivation. I personally hate motivations that boil down to "I'm doing this assholish thing so you don't have to suffer/worry, never mind that me being an asshole makes you suffer even more".

 The anime staff likely thought that letting him come back would weaken Violet's character arc.