I'm glad it has been resolved!
I'm sorry to see that you are experiencing issues with BoldGrid Backup Plugin and the restoration of your website on one domain and host to another domain and host. I recommend enabling debugging in WordPress as a first step to diagnosing the issue. While debugging is temporarily enabled, you can replicate the restore and take note of any errors that display.
I suspect since you are moving the website from one host to another, the server (specifically PHP) settings may not be the exact same. The errors that are reported when debugging is temporarily enabled should shed some light on where you can continue to troubleshoot the issue or tweaks to PHP settings you may need. I hope this helps!
The menu items' font sizes are controlled by the Theme's CSS itself. Unfortunately, this is not something that can be modified using the Customizer.
However, depending on your experience with CSS, you may be able to utilize the Customizer to overwrite the CSS manually by screen size.
I recommend trying to upload the Media file again. From the details you have provided, it seems as though the media links are broken. Did you recently move the website/files?
It appears that you are previewing the website from inside the Customizer. I recommend previewing the website from mobile as it is published since the preview will be different without the BoldGrid Customizer view. For instance, the editing buttons will not be present and the responsiveness of the theme will behave more appropriately.
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