Can she? During cases, she usually doesn't. She even admits in one of the cases that deduction is not her thing and that this way of thinking does not suit her. She helped the kids figure Kobayashi's riddle with the help of Conan's hint and she partly figured out that one case when she made the childish mischievious smile. On the other hand, Ran solved an entire case on her own, without any deduction help from Shinichi. That's not her standard, but it is possible for her. Both are not on his level, but both can keep up.
So can Ran, and so does Ran. She does it since they met each other, she admitted that much to Ran and Sera during the pre-school flashback.
Ran is more independent than any other teenager in the show. She solely runs a three person household. But you probably meant "emotionally independent". Both Sera and Haibara commend her emotional strength in the series. She hasn't cried because Shinichi is gone for a long, long time now. Last time was File 379. She doing pretty well considering that Shinichi vanished and no person knows where he is.
Again, you can't compare "completely disappearing all of a sudden" with "doesn't check in everyday". Imagine if Shinichi and Shiho walked home from school together and Shinichi says that he has something to do and runs off. And then he's gone for a few days before he calls, yet still he won't reappear and no one knows where he is (aka the situation from the beginning of the series.). You think Shiho would just say "Cool, whatever?". No way, of course she would be worried, just like Ran, just like every reasonable person would. Shinichi went off to solve cases without Ran in the past and never did we get any indication that she worried as much as she did in the beginning of the series.
No, he wouldn't. It wouldn't make any sense for him to still run after them. If he retains his knowledge, he would act completely differently, to get a better outcome for him. The more logical thing for him to do would be to tell Megure and the other policeman who are still nearby what's going down, which would have more benefits for both him and the fight against the BO. Shinichi is mostly about logic and still following them wouldn't be logical at all.