For example, in the sample sentence,
もう私は大人だから、そうは考えない.
Shouldn't second half be "そう考えない"?
Yes, it's an adverb. You can confirm with a Japanese dictionary that it's a 副詞.
https://www.weblio.jp/content/%E7%84%B6%E3%81%86
I'm not going to comment on the example sentences here. I wouldn't bet money on any of them being good examples.
Is the example sentence grammatically incorrect? Or is it valid to use は?
No it isn't incorrect. は has different usages, for example contrasting.
It's an adverb and a noun. Just like temporal nouns, e.g.:
今日雨が降る
こう雨が降る
そうする = to do that way. So そう modified the verb, it's an adverb.
そうです = it's that. Here そう is a noun.
Is そう really an adverb?
Is そう really an adverb? It seems to behave more as a noun.