As I open more tabs, more entries appear in task manager and memory use continues to grow until, finally, everything- including pointer movement slows or stops.
This sounds like two separate problems.
First, every tab is separate process and opening additional tabs does increase memory usage, this is same in all chromium based browsers. For me, 20-40 tabs have no effect in system performance however.
Second, slowing or stopping everything during this process may not related to memory usage at all - such behavior is more like related to some conflict between Opera and Windows graphic drivers (or other hardware). I have read of such problems with video-heavy sites; sometimes windows graphic drivers upgrade has helped, sometimes disabling hardware acceleration has helped - but there seems no universal cure exists.
Why closing tabs doesn't improve situation - again no answer, but I would think that some component outside of Opera control doesn't release resources. Besides graphic subsystem problems - are you using some third-party antivirus suite? Sometimes they do not like Opera cache for example.