diasy-chain surge protector with NON-surge protecting strip....consequences?

Can you post a reference to what you are talking about?
Sure, you can do that. Plugging it in that way doesn't create a problem, but you might want to re-think using one of those cheap cubes.
The cheap cube may be a bad idea. Forget surges - some of them look like fragile pieces of junk.
It doesn't matter if it is plugged into the wall receptacle or into a power strip, it will behave the same.
You may have an underlying assumption that one of those cheap cubes offers a lot of protection against surges. Perhaps against some surges they offer some protection - it migh even be sufficient to prevent damage - but to think you are well protected with one of them would be a mistake.
Ed