We can identify several views regarding the suffering of Jesus. (1) That he did feel fear and grief, but did not sin. (2) That he did not feel fear, per se, but did feel grief. Within (2), there are differing understandings of why Jesus felt grief and agony. Those who (a) accept penal substitution will argue that he was suffering on behalf of his people, while others (b) say that his suffering was due to people's sin and their rejection of him.
He did feel fear
Some argue that Christ felt fear, but that he did not thereby sin. For example, John Calvin:
[W]hen Christ was distressed by grief and fear, he did not rise against God, but continued to be regulated by the true rule of moderation. [...] Let us, therefore, attend to this distinction, that Christ, amidst fear and sadness, was weak without any taint of sin; but that all our affections are sinful, because they rise to an extravagant height.
Calvin saw two reasons for Christ's fear: the wrath of God against sin, and the physical suffering and death of the cross. Quoting Cyril,
[Jesus] allows the flesh to feel what belongs to it, and, therefore, being truly a man, he trembles at death, when it is now at the door.
Regarding 2 Timothy 1:7, Calvin argues that the command there is against cowardice, not against the feeling of fear:
[H]ere he speaks particularly about ministers, and exhorts them, in the person of Timothy, to arouse themselves actively to deeds of valor; because God does not wish them to perform their office in a cold and lifeless manner, but to press forward powerfully, relying on the efficacy of the Spirit.
He felt only grief, not fear
In this camp we begin with (a) those who hold to penal substitution, like Adam Clarke, a Methodist (Arminian). He is more uncomfortable than Calvin with the idea of Jesus being "afraid" of death:
To say that all this was occasioned by the fear he had of the ignominious death which he was about to die confutes itself – for this would not only rob him of his divinity, for which purpose it is brought, but it deprives him of all excellency, and even of manhood itself.
Instead, he suffered agony and distress because it was necessary to bring mankind to God:
His agony and distress can receive no consistent explication but on this ground – He Suffered, the Just for the Unjust, that he might Bring us to God. O glorious truth! O infinitely meritorious suffering!
Like Calvin, Clarke understands this suffering in Gethsemane in the context of the doctrine of penal substitution, which holds that Christ suffered God's wrath on behalf of his people, as summarized in the Heidelberg Catechism (Answer 37):
[D]uring his whole life on earth, but especially at the end, Christ sustained in body and soul the wrath of God against the sin of the whole human race.
Those who (b) reject penal substitution include Roman Catholics like Haydock. He holds that Jesus's grief here was not due to fear, but to the sin and rejection of men. He quotes Jerome's explanation:
The cause of our Lord's grief was not the fear of suffering; since he took upon himself human nature, to suffer and to die for us; but the cause of his grief was the unhappy state of Judas, the scandal his disciples would take at his passion, the reprobation of the Jewish nation, and the destruction of the miserable Jerusalem. Our Lord also suffered himself to be thus dejected, to convince the world of the truth and reality of his human nature.
Here, the grief is seen as a human response to sin, and as something necessary to show his human nature, but not an example of him bearing God's wrath against sin.