The keepass application can perform "auto-type" which works for all sensible applications and websites that have username/password input fields and a log-in button.
Recently, more and more websites split the log-in into two screens, first email and then password. This completely breaks auto-type and is horrible in every way. Please don't do it.
You are right that this is a good approach for many it will certainly break for many as well.
Re a) https://keeweb.info/ toss this onto any ol' free tier web host you want. No app install necessary. It's not as nice as the apps, but it works.
Re b) Is there an environment that both has a web browser that you want password management with and doesn't let you access any consumer cloud sync service?
Yes, which is why I posted the alternative to installing an app. You can use Keepass + drive/dropbox sync without installing anything using keeweb.
You do not need to install apps to access drive, dropbox, etc...
I mean, installing browser extensions to deliberately get around their security measures seems a little bit counterproductive. They aren't more secure than local apps. Do you take this company's security measures seriously or is it just some hurdle to get around for you?