My question is, if I use the public Galaxy server interface to TopHat and Cufflinks, is there any access to "cuffmerge"?

No, Cuffmerge is not available in Galaxy.

Also, I'm trying to understand the difference between using cuffmerge and then using cuffcompare (without a reference genome) to assemble gtf transcript files produced by Cufflinks for each group of 3 Illumina paired-end reads corresponding to biological replicates, in order to use the resulting combined gtf file for comparing the TopHat alignments of two such groups using cuffdiff.  

Is there any difference in the output between cuffdiff and cuffcompare, using in this fashion?  For example, do they form the "union" of transcripts by the same rules, and do their outputs contain (or lack) the same columns (strand, perhaps??)  I've read things on seq-answers indicating that I should be using cuffmerge, but I can't find it on the public server and apparently haven't installed it properly on my own computer so far.

From the Cufflinks/compare/merge/diff documentation ( ): 

*Cuffmerge calls Cuffcompare and does some filtering of transfrags as well as merging of novel and known isoforms;
*"The main purpose of this script is to make it easier to make an assembly GTF file suitable for use with Cuffdiff."

Hence, it appears that Cuffmerge and Cuffcompare are relatively similar and use the same basic union algorithm--whatever Cuffcompare uses. If you have more detailed questioned, you might ask the Cufflinks' authors:

Good luck,