Well, I would approach this like...
Do you have any natural range that gives you a relative unique set of rows which are relevant in all eight queries?
Well, in order to help you we need your queries and table preferences preferrable together with column statistics, i.e. used for ranges/unique values and so on.
How many rows do you fit per data page, how many rows are SITE_ID per SITE_ID?
Is the clustered index (table) internally/externally defragmented? I.e avg_fragmentation in sys.dm_db_index_physical_stats() above 30 percent/datafile fragmented on the disk.
For example the natural range could be SITE_ID AND ACTION, or SITE_ID AND USER_NAME, or ORG_NAME AND ACTION, or what ever... Right now we're only guessing.
If you can see you natural range, in relatively small sets of rows, fitting a single data page, or a UNIFORM extent, while meeting your need of returning rows and all eight queries, try that för a CLUSTERED INDEX.
Or you might even consider creating multible non-clustered indexes with included columns.
CLUSTERED ON (SITE_ID, ORG_NAME) -- Do they express the same thing?
Qry1/Qry2/Qry3/Qry4, Qry8: ON (USER_NAME) INCLUDE (ACTION, ACTIVITY_NAME, FROM_PAGE, LOG_DATE_TIME, SESSION_ID)
Qry5/Qry6/Qry7: ON (ACTION) INCLUDE (ACTIVITY_NAME, LOG_DATE_TIME)
But then again, just guessing right now...