Right, that's something that always bugged me too. It's just.. Assumed that "welp, I guess I'm evil now and I'm serving this guy who fed me lies, ruined my life and failed to deliver on the one promise that I used to betray the good guys". It's just never addressed.. Ever
He's got no reason to follow the emperor.. To not jump on the nearest ship and get outta town.
Literally everything was hinged on Padme. His new powers, empire, world he wanted to build.. It all rested on the assumption that Palp had a way to save her from death.
Um, it makes perfect sense. First, as already stated Anakin was already full of hatred regarding all aspects of Jedi life, so he didn't just turn to the Dark Side for Padme. He had other reasons, but Padme was the final push. Second, Palpatine was the only one willing to help him with Padme. He never directly stated that he knew the secret to saving Padme's life, only that he was the only one that could help him do it. Once he offed Windu, Anakin had to stick with Palpatine as he was the only one who could possibly gain him the knowledge he needed. Also, as stated before, joining Palpatine meant being able to finally live in freedom. He could finally openly express his love for Padme and not hide it like a dirty secret. Third, Anakin always wanted power and Palpatine was going to give him ultimate power by switching sides. Which ties to the last point, Anakin was already willing to betray Palpatine and rule the galaxy himself alongside Padme. So, it's not like he was some blind slave in his mind, he was fully ready to ice Palpatine if he couldn't offer him anything and just take everything over himself.
As for the buzz droids, all I can say is that I personally like the whole sequence and think it fits perfectly with the tone of the opening. I think it's silly to try and argue which goofy sequence in SW are better than the others. I'd rather watch the buzz droids ten times over than watch Finn and Rose ride through Canto Bight on CGI horses. Goofiness is goofiness, it's pretty much impossible to argue which is better, you're just advocating for your personal taste of goofy nonsense.