Hi there,
Let me clarify some things which might help you.
If you've switched to using a local account on your laptop and have signed out of your Microsoft account, it will no longer sync with the Microsoft account. Syncing occurs only when you are actively signed into your Microsoft account on the device. So, if you are now using a local account, the laptop won't be interacting with your Microsoft account.
As for the impact on your laptop and settings when the Microsoft account is closed, any services tied to that account will cease to function for you. For example, you won't be able to use Microsoft's suite of services like Microsoft Office Online, OneDrive, Outlook.com email, Xbox Live, or the Windows Store. However, the basic functionality of your laptop which doesn't require a Microsoft account will continue to operate as it does now. Your local files, apps, and settings should be unaffected.
Renaming your laptop won't have any effect on the link between your laptop and your Microsoft account. This linking is done through unique identifiers within the Windows system, and not your device's name. It doesn't matter if the device's name changes, the system still has record of that device having been linked to the account.
It sounds like your main concern is being unable to unlink your device from your Microsoft account due to having lost your password. If that's the case, you could attempt to recover your Microsoft account password. You could do this by visiting the Microsoft password recovery page:
https://account.live.com/password/reset
Here, you will be guided through steps to prove your identity and recover access to your account. If successful you can then formally remove the device from your Microsoft account before closing it.
I hope this helps clarify things for you! If you have any more questions, don't hesitate to ask.