Re: The story about the Satmar Rav, I have also heard the story just that it involved Reb Moshe, not Rav Aharon Kotler. It has nothing to do, however, with the Satmar Rebbe’s minhag per se. In Satmar, as in other branches of Chassidus such as Tzanz and Amshinov – the minhag is to hold Rabbeinu Tam k’din, meaning – that they hold both shkia and tzeis according to Rabbeinu Tam.
The way I heard the above-mentioned story is that the reason Reb Moshe asked the Satmar Rebbe to stop driving after the earlier shkia is because Shemiras Shabbos in America was a major problem and he was choshesh that people seeing a gadol driving after shkia would complicate things further.
While today one doesn’t see many Chassidim doing melachah after the earlier shkia, those who hold Rabbeinu Tam k’din will wash for Shalosh Seudos, daven Minchah, etc. after the earlier shkia yet before Rabbeinu Tam’s shkia. The converse is also true – they will not do melachah or break a fast before Rabbeinu Tam’s tzeis.
Anyone who wishes to pick a bone with Rabbeinu Tam is welcome. But don’t tell Chassidim that they are being oveir aveiros because you hold differently.