The only two I can think of are the Articles of the Confederation and Hong Kong. What are some others?
I wouldn’t say the Constitution intended for a minimal state. It needs to be destroyed.
No and no. Under the Articles the individual states can do pratically anything. Hong Kong is anything but minarchistic. Iceland is perhaps the most limited state, in that the number of PDAs were limited to 39, but no other statist restrictions existed.
I think you jumped the gun a little too quick there. 
OP asked for the most minimal state, not the most minarchist state (whether or not that can/has exist(ed) is an issue for a separate thread).
So it’s all about relativity. How does Hong Kong not sit towards the “relatively” smaller state side?
European microstates. Those and small (geographically) states in general.
Speaking of micro-… http://en.wikipedia.org/wiki/Aerican_Empire
The Scandinavian states in general had some pretty small governments for a long time. Now that changed, of course, but it is (part of) the reason, why they are still quite well off.
Building on what Peter Sidor said, check out pre-socialist Sweden (about ~ 1930). Very laissez-faire country, with a quiet foreign policy (except norway).