IRULE for redirect request (http to https), append request and rewrite response (http to https)

For 1&2, you can use an httpclass (pre v11.4) or a policy (11.4+) to do that, no iRule necessary.

For 3, google would need to rewrite that ajax call, the request isn't coming to your domain. However, assuming it's your app that is inserting that, you can use a stream profile to rewrite that URL to ssl. You can try this (untested, check out and tune as necessary in lab environment)

when HTTP_RESPONSE {
  STREAM::disable
  set stream_expression "";
  if { ( ([HTTP::header Content-Type] starts_with "text/html") && ([HTTP::status] == 200) } {
    set stream_expression "@http://www.youtube.com@@"
  }
  if { "" ne $stream_expression } {
    STREAM::expression $stream_expression
    STREAM::enable
  }
}
when STREAM_MATCHED {
  if { [string tolower [STREAM::match]] eq “http://www.youtube.com” } {
    STREAM::replace “https://www.youtube.com”
  }
}

give this a shot:

ltm policy http.redirects {
    controls { forwarding }
    requires { http }
    rules {
        redirect_rules {
            actions {
                0 {
                    http-reply
                    redirect
                    location https://www.abc.com/test/login.aspx
                }
            }
            conditions {
                0 {
                    http-host
                    host
                    values { www.abc.com }
                }
                1 {
                    http-uri
                    values { / }
                }
            }
            ordinal 1
        }
    }
    strategy all-match
}