Is it permissible for a soldier to pray without ablution (wudu'), or with a contaminated military boot, or in a hasty manner due to the nature of his work?

For the prayer to be valid, purity from both minor and major ritual impurity is required, either by water or by tayammum if water is unavailable or its use is impractical. Prayer without wudu or tayammum, when they are possible, is invalid, and doing so intentionally is a great sin.

As for praying in shoes, it is permissible. The Prophet (peace be upon him) prayed both barefoot and in shoes. In fact, it may even be desirable to do so, to differ from the Jews, after ensuring their cleanliness. If there is any filth or dirt on them, they should be wiped on the ground, and then one may pray in them.

As for praying quickly, it is not valid if it violates the pillars of prayer.