Let me tackle your title part by part.
Directly? No. HOME is server based, and PKHeX is a save hacking tool.
Indirectly: Place the mons in your Sword save, then place them in HOME.
Yes, just modify your Pokemon Sword save in PKHeX. Requires a CFW'd switch tho.
I already answered this: cannot directly go into HOME.
At present, there's no DNS trick public to accomplish this.
Here's a suggestion, and I hope you can follow this train of thought:
Nintendo Switch allows you to create multiple user profiles.
Now, assume user profile A has your "legit" save.
Create a user profile B. Create a Sword game save using B (as in start the game then save...)
Use a save manager to dump the "legit" save in A (let's call this A save).
Make copy of A save (named A save copy now), THEN edit it with PKHeX, then save over A save copy.
(this means you'll have 2 saves: A save and A save copy)
Now inject A save copy into B. (Now's let's call this B save)
Now, HOME allows you to view saves for other user profiles.
So, starting up HOME with A, use it to view B save.
Drag the hexed mons into HOME.
Then still using HOME, access A save.
Now drag the hexed mons into A save.
Congratulations, you have edited mons in A, without editing the A save.