Can you find out when a trojan condom was manufactured?

Probably.

You'd have to contact the company with the serial number on the box, and probably go through a whole bunch of automated hoops, but you'd probably get it by contacting the company.

Yes, and look on the box or wrapping to see if there is a 1-800 phone number and call it to find the manufacture date or the "freshness date".

I once called the 1-800 number on a can of tuna and gave them the numbers stamped on the top of the can, and they gave me the date.

Do you have an ultraviolet light handy? Really. TAke the box or wrapping into a dark space and slowly scan over it with a UV light. The date of manufacture and other numbers will glow under the UV.