When Paul uses “we” here, he is referring to himself and Timothy. (See: rc://en/ta/man/translate/figs-exclusive)
“We prove that we are God’s servants by all that we do”
Paul mentions various difficult situations in which they proved that they are God’s servants.
Paul lists several moral virtues that they maintained in difficult situations that proved that they are God’s servants.
Their dedication to preach the gospel in the power of God proves that they are God’s servants.
“Word of truth” here is a metonym for “message about truth.” AT: “in the message about truth” (See: rc://en/ta/man/translate/figs-metonymy)
Paul speaks of their righteousness as if it is weapons that they use to fight spiritual battles. (See: rc://en/ta/man/translate/figs-metaphor)
“righteousness as our armor” or “righteousness as our weapons”
Possible meanings are 1) that there is a weapon in one hand and a shield in the other or 2) that they are completely equipped for battle, able to fend off attacks from any direction.