Nobody wants to be a micromanaging coworker. It's one thing to be a micromanaging manager. It's not an ideal management style, but at least their employees are managed by the person who is supposed to manage them, not by someone else. When you're an equal to your coworker, you should not be going around telling them what to do about basically anything. If they smell bad, you should tell HR so they can address it professionally. If they're just plain annoying, there is very little you can say in the workplace that can actually fix that problem. If they're bad at their job, it's not like you can fire them! You might feel uncomfortable telling your manager that they aren't picking up the slack, because maybe that's not your call to make. The best thing you can do is just be friendly with your coworker and hope that they aren't a total disaster.
Sometimes there are things that need to be addressed with your coworker that aren't as easy as asking them to pick up a shift for you. They might be saying a lot of unprofessional things that make the workplace uncomfortable. They might talk so much about unimportant things in your open-concept office that it's impossible for you to focus on the work that actually needs to get done. In cases like these, I can't emphasize enough how important it is to talk to HR. I know people say HR is there to protect the company, not the employees, but I highly doubt an HR professional will fire someone for mentioning a coworker who smells so bad they can't focus on work. If anything, they're probably not the only ones complaining about this issue if it's as bad as they say. The employee in this story desperately needs to talk to HR about his coworker, who wants him to stop washing his hands with scented soap because she is sensitive to the smell.