Tell us what you are trying to do?
In the settings of Toolset, I added a few custom codes which I got from the technical support team. Is there any way to export and import them?
If you added the custom code at Toolset > Settings > Custom Code then that adds the code in individual .php files in a folder wp-content/toolset-customizations/.
To use the same snippets on another site you can simply copy those .php files to the same folder on the other site (if the folder doesn't exist—because you didn't create any custom code yet—you can first add it).
The snippets won't be active until you activate them.
I can see that there is an index.php file in toolset-customizations folder. If I am transferring the snippets from my old site to new site and I don't have that toolset-customizations folder, do I need to move the index.php file too?
Yes - you can just copy the "toolset-customizations" folder from your old site to new site.
I suggest you should create code snippets first on new site with the same name as old site and then copy all the .php files. Make sure the slugs of the code snippet should be equal to the .php file name.