compare di-smiss, dis-miss?
Also cp Schmiss "trash, stuff", Schmeiß-Fliege "drosophila" (dew lowe, a.k.a. house fly), and schmieren, English to smear, smear campaign "deface, denigrate"; given dismiss cp to mete and I don't know what; also niederschmettern, to smite, abschmettern (of law suits).
Whichever way it went, rausschmeißen is chiefly denigrating.
rauswerfen, verwerfen, etc are not exactly positively connotated, but slightly less derogatory. Perhaps it's distantly relatrd to rufen, cp in Verruf geraten, opposite of Beruf and guter Ruf; That is, if the basic root, *weH meant to turn, change, the apparent relation to werfen and warp might be coincidental.
Either invokes the visual metaphor of hurling someone, e.g. out of the saloon.
As in English (e.g. "to let go"), euphemisms are usually sought to maintain politeness, kündigen, etw. entziehen, bitten zu gehen, ...