'Just' is singular, 'unjust' plural. There is no article in either case. It is not 'the just' par excellence, as Acts 3.14.
The article being left out, it is characteristic, in contrast with 'in flesh.' Both flesh and spirit are the manner and character of what is predicated of Christ. We could say 'present in spirit,' 'fervent in spirit,' because it is characteristic: but 'made alive in spirit' conveys to the English mind the idea of an accomplished fact. It cannot be simply characteristic. In Greek, on the other hand, although conveying a fact, it has a characteristic significance. The sense given here is right.
Or 'disbelieving:' see John 3.36 and ch. 2.7,8.
The Greek means 'arrive safe into a place of security through difficulty or danger,' as Acts 27.44.
This does not mean, I think, that they went through the water to get in, i.e. through the course of the flood. The apostle's mind does not turn to the flood, but to the water as an instrument. Water was ruin and death, and they were saved through it.
Or 'engagement,' or 'testimony.' The Greek word here translated 'demand' is a very difficult one, and has puzzled all critics and commentators. It means 'a question.' All the commentators speak of its use as a legal term with the sense of contract, or rather stipulations or obligations of a contract. I judge (as usual in these forms) that it refers to the question asked rather than to the asking of the question. The legal use arises from a questioning which settled the terms of the contract, hence called the questioning. I am disposed to think it is the thing demanded. It requires as before God, and has it in baptism as a figure by the resurrection of Jesus Christ. It must be remembered that 'of a good conscience' in English may be the thing requested or 'he who requests.' The form of the word in the Greek would rather make it the thing requested or demanded.