No one bothered to question his age. If a man seven feet high says he's eighteen, people tend to believe him, whether he's started shaving yet or not. And he was pretty fit. Hagrid had always been an outdoorsy person, and tramping around the wild had kept him in good condition.
The army were pleased to have him.
Hagrid didn't take much with him. He didn't think carrying his old bits of wand around would be safe, so he reluctantly left them at his father's house.
Hagrid quite enjoyed parts of army life – at times, it was almost like school. You had your group of friends, you did as you were told, and you were rude about your superiors when their backs were turned. The beds were too short, as were his trousers, but that was something he'd learnt to live with before. The other soldiers called him 'Titch'. Apparently this was some sort of muggle joke.
He quite liked his gun, too. He carried it around and took good care of it in much the same way as his wand, and he became quite attached to it.
Right up until the moment he had to use it to kill someone.