Are weather-reporting requirements for an airport in Class D airspace more stringent than for an airport in surface-level Class E airspace?
Your First Question: In the US, are weather-reporting requirements for an airport in Class D airspace more stringent than for an airport in surface-level Class E airspace?
Answer: The &quot;Primary&quot; airport within active Class D airspace requires (Ref: FAA Order JO 7400.2M, paragraphs 17-2-9 and 17-2-10:

That a Control Tower be currently in operation;
That a certified and approved weather observation and reporting system (for the primary airport) be in place and operational; and
Communications capability must exist down to the runway surface area (for the primary airport).

However, a primary airport that is within the surface area of active Class E airspace will not have a Control Tower currently in operation.  (if there are airports with a currently operating Control Tower within the surface area of active Class E airspace, I am unaware of them).
With respect to your specific question above, if the airport in question (within the surface area of active Class E airspace) is meant to accommodate instrument procedures (e.g., instrument approaches, IFR arrival/departure procedures, etc.) then the exact same requirements (Ref: FAA JO 7400.2M, paragraph 18-1-3) for communication and weather observation/reporting must be available as are required for Class D surface airspace. (But there will not be a Control Tower operating)
There are a dwindling few airports that have Control Towers that operate while the airspace is designated as active Class G. But, as I mentioned above, the same is not true for primary airports located within the active surface area of Class E airspace. In that case, the Control Tower, if there is one, is not operational (e.g., closed/not staffed - usually in the later evening as traffic warrants).  If the Control Tower was currently operating the active surface airspace would be Class D.
Your Second Question: Are there other requirements that would cause an airport with an operating control tower to be eligible for surface-level Class E airspace but not for Class D airspace?
Answer: Again, as mentioned above, if the primary airport has a currently operating Control Tower, the active surface airpace would be Class D (or in a few cases Class G), not Class E. Therefore, parsing the eligibility requirements between Class E and Class D airspace, both of which (with respect to your question) might have operating Control Towers, is not possible.
Typically, when/if the Control Tower closes (not staffed/closed for the night) at a primary airport in Class D surface airspace, then that airspace surface area reverts to Class G or Class E.
Lastly, my response above excludes special occasions where a temporary Control Tower may be operating in active surface airspace designated as Class E.  This would be a circumstance where a NOTAM or similar would exist identifying a particular event taking place.