Just a few additions and minor corrections.
Seitdem (a conjunction and an adverb) and ohnedies (an adverb) have a pronoun (and a case) embedded. That's why they are different from "a lot of prepositions that take the accusative or the dative". Similar: trotzdem. (Even though most speakers use the genitive case after "trotz". Still, they wouldn't use "trotz dessen", angesichts the fact that there is the convenient contraction trotzdem. The older form "trotz + Dat" is not incorrect and still used by some writers, whereas "wegen + Dat", though in frequent use, is considered incorrect).
Furthermore, in contrast to your other examples ("wegen des Wetters.../Deswegen..."), seitdem belongs on the side of "deswegen/darum", that is, on the "answering" side. Like "deshalb" it refers to something that has been mentioned before. "Vor drei Jahren hat sie geheiratet. Seitdem spricht sie nicht mehr mit ihren alten Freunden." "Seitdem" refers to the whole of the sentence before as does "deshalb" in "Es regnet heute fürchterlich, deshalb gehe ich nicht ins Büro." -- The conjunction "seitdem" is equivalent to "seit" and stylistically inferior as it introduces an unnecessary duplication. "Sie spricht nicht mehr mit ihren alten Freunden, seit(dem) sie geheiratet hat."
Entgegen takes the dative case. If you refer to it, pronounwise, you say dem entgegen. Compare the contractions dementsprechend and demgemäß.
Angesichts does not give a reason (unlike wegen or "[Gen.] ...halber"). In other words, it doesn't answer to weswegen, weshalb, warum?. That's why a substitution with "deshalb, darum..." doesn't work. As an answer to "why did you stay at home?", angesichts des Wetters would not be an idiomatic reply, whereas angesichts des Wetters bin ich lieber im Bett geblieben would at least allow you to infer a reason: while "angesichts" is neutral in itself, this reply as a whole makes it clear that consideration of the weather in this case means "bad weather", so you understand it as "because of the bad weather". Unlike wegen or ...halber, angesichts does not specify a cause but merely describes circumstances.
@#2,3: No, I don't think that "In Angesicht" works (and not only in Switzerland). There is "im Angesicht des Sturms..." (confronted by/with the storm, facing the storm, being reduced to (his/her/one's) own abysmal existence, eye-in-eye with the unwordly threat of the storm... -- whatever fits) and "angesichts des Sturms" (taking the storm into consideration, having taken notice of the storm etc.).
Halber is tricky, as it is an old declension of halb as a substantive, meaning "Hälfte" or "Seite" (Duden tells me). I can see Rex's point in #10 (it seems that halber is frequently used in the sense of "for the sake of", "der Bequemlichkeit halber" = "um der Bequemlichkeit willen" = for the sake of convenience, giving in to... etc.) but don't find anything wrong with "des Wetters halber bin ich nicht ins Büro gegangen". While Duden remains silent on this point, Brockhaus/Wahrig gives both meanings: wegen... and um ... willen.
Sorry for getting so effusive. And bix's #11 was not OT but necessary. And this was definitely a question for Language Lab. And ... over.