You do always need to check in and to check out at the same company. That is how the system works. Checkin means a valid card, checkout means paying the right amount for your journey to that company. Otherwise you are in trouble and there will be an extra fee.

With the train you can correct an error with a wrong checkin right away by checking out within an hour. With other public transport you will pay a small fee when you checkout right away,

You can always check what has been charegd in mijnNS-if you have subsciption-or via the screen on machine for the blue anon. cards. But it is likely you did wrong and thus were charged extra-this is also shown in amount left-as 10 or 20€ will then be charged. You can correct this via site uitcheckgemist.
It would have been helpful to tell what kind of card/subscription you ahve

If this was at e.g. Amsterdam Amstel, it should not give you a double check-in but give you 3 alternating beeps, a small red light and a warning on the screen that you're already checked in.

Theoretically (but unlikely) it could result in a check-out, after which you pay the boarding rate (€ 4,00) at your destination because of a missing check-in. See www.uitcheckgemist.nl to correct that.

You can check what happened with a Mijn OV-chip account https://www.ov-chipkaart.nl/ov-chip-and-use/use/how-does-my-travel-history-work.htm

The other way around works the same, when checking out at the platform and then going through the gates, they will simply open and say ‘Goodbye’ without another check-out.

No, when checked in at the gates one will be checked out after using one of the poles on the platform. The other way around (checking out on the platform and then passing through the gates) indeed works as described by you, as from the inside of gates you're only able to check out again.

Ah, in that case see uitcheckgemist.nl to correct the missing check-in. You will get part of the € 4,00 back.