Ok, when you have "---0---/8---7" you are supposed to hit the E note and without hitting the string again kind of lightly hammer your finger while sliding up to the C note (8 fret). Although it is quite simple, it might sound wrong if you hit the slide too hard, if you listen to the Stairway Recording you will hear Page play that section very "soft". As far as I can see from your post, no, you should not pluck the string twice. Also, I have the transcribed series of Led Zeppelins recording and although to me it sounds better to play it as you just tabbed it, it appears like this: "---0---6/8---7---" It also tells you to accent the slide, just in case, to accent means to kind of make it sound more noticeable than the other notes in that section, it could be, for example playing it more louder than normal...
Hope this helps you out a little bit because you will see that same technique in a lot of other songs like Hendrix's and Rhoads...