irule processing and performance of 2 irules both doing http_request

There isn't much you could save by trying to combine the iRules or share variables across them. However, you can optimize both rules separately:

 logging iRule
when CLIENT_ACCEPTED {
   set hsl [HSL::open -proto UDP -pool pool_syslog_514]
}
when HTTP_REQUEST {
    Save the request components we want to log in the response
   set log_string <190> src_ip=[IP::client_addr], dest_vip=[HTTP::host],http_method=[HTTP::method],\
      http_uri=[HTTP::uri], http_version=HTTP/[HTTP::version]"
}
when HTTP_RESPONSE {
   HSL::send $hsl "$log_string, return_code=[HTTP::status]"
}

Pool selection iRule

when HTTP_REQUEST {   
   switch -glob [HTTP::path] {
      "/123/help*" {
         poolserverpool01_8080
         persist none
      }
      "/123/petsemail*" {
         poolserverpool02_8080
         persist none
      }
      "/123/profileeditor*" {
         poolserverpool03_8080
         persist none
      }
      "/123/activitystream*" {
         poolserverpool04_8080
         persist none
      }
      "/123/core*" {
         poolserverpool05_8080
         persist none
      }
      "/123/*" {
         poolserverpool06_8080
         persist none
      }
      "/admin*" {
         if { [IP::addr [IP::client_addr] equals 10.1.1.1] or \
            [IP::addr [IP::client_addr] equals 10.50.1.1 or \
            [IP::addr [IP::client_addr] equals 10.100.1.1 } {
            poolpool_qlaxorin_8080
         } else {
            HTTP::respond 404 content { 404 Not Found       }
         }
      }
      default {
         HTTP::respond 404 content { 404 Not Found       }
      }
   }
}

I haven't tested the iRules, but hopefully they give you an idea of how you can make these more efficient.

Also, you could add the hosts/networks you want to allow access to the /admin URIs from to an address type datagroup and then use the class command to do the lookup.

Aaron