Stapled old and new passports

I would suggest you separate them. While stapling them together is "allowed" (the old passport is expired and invalid anyway, and staple marks on the back cover of a new passport are not material damage), there is generally no point to submitting old passports, since records are electronic these days.

The only reason you might need an old passport is if it contains valid visas and the country you are going to accepts visas in expired passports, but it doesn't sound like this is the case. And even if it was, there's still no need to physically staple them together.