That's essentially what you do when you wipe your system and install another distro. If you have a separate home partition that stays intact through the process then it's especially true.
Technicly yes but it'd be hard and would probably break a lot of stuff. If you need a diffrent package manager its better to just install a distro that comes with it.
curious, why would you want that?
the packages are usually built by the distro maintainers for the distro, so if you switch from, say apt to dnf you'd have to take care of repackaging yourself.
Well, technically ya could. Given some effort you could entirely replace it, but there's usually no point. Take NixOS' package manager as an example. You can drop it into any dsitro and use their repos, but anything you can find there you'd usually be able to find in your native repos.