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Of course it does. Open source projects have a decent chance of being included in your distribution's default repositories. Popular packages will even usually be kept up to date. All that for free, with no effort on your part.

My experience tends to be that distro packaged software is not always what one ends up using in production. Except for some base packages when you start doing more sophisticated deployments. Distros may backport bug fixes and specifically security fixes but they tend to be several versions behind.

It does has a bearing on the ability to migrate to an alternative if it turns out to become awful, because someone will probably write an export script in that case.

It's one of the big advantages of FOSS: no vendor lock-in.

Yes, it does.

A software being open source means that you can deploy for any task in any environment, at any number, without the need to ask anybody's permission.

It being proprietary means that you must ask permission first, and will probably need to keep each piece of software under control, under different terms.