I've got the latest LimeWire for Mac, and it's pretty buggy. After maybe ten minutes of running -- and like thirty searches for various groups or songs -- the thing stops searching for music. For another minute or two it'll let me do new searches, but it won't pull up any results for any of them, and it won't switch from that "search" screen to the "summary" screen. After that, it won't accept any new searches at all -- hitting the search button does NOTHING.

I'm thinking maybe this has something to do with the memory I allocated LimeWire. Is 64000K minimum and preferred enough? Should I go for more? Or is there another reason LimeWire won't accept more than a finite (and SMALL) number of searches before I have to restart?

I'm presuming you're talking about OS 8 or 9 right? You didn't specify so it's hard to know what operating system version you're talking about.
Make sure you have the last version of Java for OS 9: MRJ 2.2.6 for OS 9.2.2 http://docs.info.apple.com/article.html?artnum=120209 (previous Java from here MRJ 2.2.5 http://docs.info.apple.com/article.html?artnum=75097)

Some interesting reads very relevant to your problem:
1. http://www.gnutellaforums.com/showth...8221#post98221

2. http://www.gnutellaforums.com/showth...threadid=25090

I would suggest it is mostly a Java issue & the other part a memory issue. The earlier versions of Java are not memory efficient or generally efficient, & Mac is slow to update Java versions. If you want updated Java then you need to upgrade your operating system to a higher version.