This works, but if you've previously leaked a copy of the old hashes, then old passwords which are still in use are still only as secure as the old hashing mechanism. They won't be running old hashes against Dropbox's live service, they'll be dropping guessed passwords against Dropbox's live service. It doesn't matter if that password is now secured by bcrypt-with-a-hefty-cost-factor wrapped around MD5, improving its security versus being done in just MD5 -- if the password is right then the attacker will get through.
I'm only on two OSes (Windows, Linux) and predominantly two browsers (FF, Chrome), but Keepass (Works in Windows and Linux - pretty sure Mac too) + KeepassHttp plugin + PassIFox [1] + ChromeIPass [2] works for me.
I find it also nicer than LastPass as it combines with the native password saving of FF/Chrome rather than the highjacking and restyling of input boxes that look like user/passwords done by LastPass.
It's probably fine so long as both your 'client' and 'server' are on localhost, but it'll probably never be secure if the server is exposed to the internet.
I regularly switch between a MacBook for work, Linux desktop at work, Windows 10 home machine, iMac home machine, iPad, and iPhone, using at least two browsers on each (Chrome/Safari or Chrome/Edge) and often two Chrome profiles.
I use 1Password with the encrypted vault synced to Dropbox and it works wonderfully well. 1Password has native apps / browser extensions for all of those platforms (except Linux local app where they support and recommend using Wine).
I use KeePass with Keepass2Android (which has Dropbox integration), and Firefox Sync. You can log in to Firefox Sync and all your passwords will be there with you, but the canonical storage is KeePass. If I'm at a resetting computer, I just type it by hand from the mobile phone. It's not quite as convenient as using one password for everything, but it's a little bit less convenience for orders of magnitude more security.
Not unless you make sure you backup your password file somewhere other than dropbox. It doesn't even have to be very often, just whenever you update your Dropbox password (for example).
