Does time logged in these foreign-registered aircraft (cross country, etc) count towards US certificates or ratings? Or is it flight experience that should be excluded from an 8710?
Yes, it counts: the FAA doesn't care where flying hours are accumulated provided that the US written and/or practical test requirements are met. I learned to fly in South Africa then applied for an unrestricted FAA license (i.e. not a foreign based license). The FSDO told me that I could count all my time in SA (cross-countries, night flights etc.) towards the FAA license requirements, but I would still have to pass the FAA written and practical tests like any other student pilot.
One practical issue in this situation is that hours may be logged slightly differently in different countries, by regulation or simply by local practice. The two examples I encountered were that PIC time is (was?) defined differently in SA and the US, and the number of landings is not relevant in SA for currency so no one logs them.
This is from FAA website:
(j) Aircraft requirements for logging flight time. For a person to log flight time, the time must be acquired in an aircraft that is identified as an aircraft under §61.5(b), and is—
(1) An aircraft of U.S. registry with either a standard or special airworthiness certificate;
(2) An aircraft of foreign registry with an airworthiness certificate that is approved by the aviation authority of a foreign country that is a Member State to the Convention on International Civil Aviation Organization;