On the title question. We can get rid of Astarion, one of the companions that will be playable in the full release, and just found out that Shadowheart too can be subtracted to the party, so I think there's a way to get rid of every and each companion.
Gotta say this post made me curious so now I want to try runs without recruting this or that companion (can not afford to try to make the character despise my toon enough or the trigger to push them to leave the camp with my playstyle I end up with neutral approval by Lae'zel and Astarion and high approval from Shadowheart, Willy, and highest approval from Gale).
On my solo run I did not talk to any of them. My journal mentioned that Lae'zel died while searching for her people, probably because I had met her on the ship. Wyll died during the fight. Gale and Astarion never showed up after I avoided their starting areas, and Shadowheart appeared at the druid encampment. I did not talk to her there and later she appeared in a rather buggy manner at my camp as if she belonged there. Dialogue implied I had already had her join. To get what was supposed to happen I reloaded from before I spoke to her and pretended she wasn't there, then on a long rest she "showed up" at my camp saying the voices told her to find me. She had her hand too close to a weapon so it ended badly for her.
May try a run where I do everything I can to get negative approval with Gale or Astarion to see if I can get them to leave, not sure if it is possible though.
You can get rid of every single one and killing all of them permanently, you can technically play the game solo but good luck with some of the fights. some of the enemies will one shot you.
You can solo, it is actually fun. Have only done it with a ranged character so far, can't decide if I want to do melee (probably Battlemaster since EK is gimped without certain spells) or magic (Abjuration Wizard probably) next. Been taking a bit of a break until another update.