I can't really help you with a, that's largely dependent on you. Whether it meets requirement b depends on how you're going to be using it, really. My use case is mostly websites, in Firefox and Chrome, and occasionally on my Android phone. I wanted the UI to integrate nicely with the browser, to be able to store other passwords besides, for the architecture to be secure (passwords are always encrypted with your master password before going to their servers, and your password is never transmitted to them), for it to have two-factor authentication, to sync automatically, and for it to have a website backup for access without the extension.

LastPass fit all these, but it might not fit your usage requirements. However, I would urge you to investigate it and all other options (KeePass seems popular, as are some other Firefox extensions) if you don't currently have a good password solution, as any of these is better than using the same password multiple places.