The sword of Revelation is not always ῥομφαία (rhomphaia). In Rev. 6:4 a different word is used, which means ‘a great sword’ – wielded by the rider on the red horse: “and power was given to him that sat thereon to take peace from the earth, and that they should kill one another: and there was given unto him a great sword.” This rider has but one symbol of death to wield – a great sword – massive compared with the short swords Roman soldiers generally used.
Of note, the rider of the pale horse has four means of death to deal with a fourth of the earth; the sword, hunger, death, and the beasts of the earth. This is reminiscent of the prophecy in Ezekiel chapter 14. God warned of his judgments on Jerusalem, sending famine, causing beasts to pass through the land to render it desolate, bringing a sword upon the land, then pestilence. “For thus saith the Lord God; How much more when I send my four sore judgments upon Jerusalem, the sword, and the famine, and the noisome beast, and the pestilence, to cut off from it man and beast? (vs. 21).
Unsurprisingly, the prophetic vision of God’s judgments upon the world in Revelation chapter 6 uses the same imagery. Now, you ask specifically about the sword, but – really – why pick on one symbol for death when other ones are used?. Certainly, the sword is referred to quite often in the whole of the Book of the Revelation, but if that writing teaches us anything, it is that all of such figures of speech are symbolic and not literal. Jesus does not literally have a sword coming out of his mouth, any more than robes become white by dipping them into the blood of the Lamb (the risen Christ). You have already stated that you know the sword coming from the mouth of Jesus is symbolic, of course, so that is understood.
In that case, your two specific questions require symbolic answers. The first one, ‘Is the sword in Rev 6:8 also from the mouth of Jesus?’ The text does not say that it is. The rider on that pale horse appears with his sword already in his hand. So, No, is the answer. You add, ‘or at least wielded on behalf of Jesus?’ The riders of the red, black, and pale horses are symbolic of what is to happen on earth after the Lamb opens the second, third, and fourth ‘seals’ (Rev. 5:1-8). These start the judgments of God, from when the glorified Christ in heaven started opening those seals, from when he rode forth to triumphantly conquer with a bow (yet no mention of arrows – Rev. 6:2). Once Christ rides forth on his white horse, the other horsemen follow. Those who acclaim Christ’s triumph of the cross and the resurrection are conquered by his love, not by force. But others who disdain Christ begin to suffer the effects of the other horsemen. In every generation, every century, large swathes of the population have been struck with various forms of violence that they symbolise. As one writer puts it:
“A lifeless corpse is pale: likewise the colour of the horse; and
Death the name of the rider, whose companion with him is called Hell.
The Lamb opens the seal, and these canter forward into time on their
grisly errand over all the earth, with power given to them from
Almighty God to kill the fourth part thereof, again and again,
generation by generation. “Death effects this and by four means, for
the earth, the whole earth, is that from which the fourth part is
reaped by the grim reaper, and meanwhile Hell drawing the souls into
its habitations until the resurrection at the Last Day… The cause is
from above; only the means is from below. But fallen man sees nothing
but the means… and three quarters of today’s people barely notice
their passing. Barely notice. Nor yet at all do they hear the
hoof-beats of the pale horse at their own heels.” The Revelation of
Jesus Christ by John Metcalfe, pp161-163
So, the rider wields death to a quarter of earth’s population as an agent of God’s judgment, is the answer to the second part of your first question.
Your second question, ‘Is the killing with the sword in Rev 6:8 related to the killing with the sword in Rev 19:15, 21?’ seems to me to be ‘No’. That is because Revelation chapter 19 has moved on in time to Christ’s return to earth in glory, with hosts of angels, to destroy all the wicked. All the angels are on white horses (vs. 14). No mention is made of red, black, or pale horses now. They have completed their grisly ride. All the horses now are white. This is when the grapes of wrath are gathered and cast into the vat, to be trampled on. This is the culmination of all the plagues of judgments that have gone on before, throughout the generations, in all the centuries from Christ’s ascension till his return. This is when the unseen satanic forces at back of all earth’s turmoil are “cast alive into a lake of fire burning with brimstone” (vs. 20). The last trump has sounded, and time is no more. The first heaven and the first earth have passed away, replaced with a new heaven and a new earth, and then comes the marriage of the Lamb. Hallelujah!