No, you should be able to clone landing pages in design studio. Must be a permissions thing, check with your administrator account holder to adjust those.
No, you should be able to clone into a program as well. When you first enter the clone interface, the first question is "Clone To:" The choices should be a) Design Studio and b) a program
I clone landing pages all the time within the program. But if you're going to do some major landing page template changes like changing the copyright info or the logo, it's best to do it from the Design Studio.
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