Yes, for Kobe you should head for "shin-kobe" shinkansen station. For a 7 day Japan Rail Pass to be worthwhile you need to spend over ¥28,300 on JR tickets. Lets take a look at your costs:
- Tokyo > Shin-kobe: ¥14270, (195mins, 589km)
- Shin-kobe > Tokyo: ¥14270, (195mins, 589km)
TOTAL: ¥28540 for 2 legs
As you can see it's very slightly cheaper to go with a 7 day rail pass than standard tickets so I can recommend this choice for maximum flexibility. For example you can go on day trips to Hiroshima or Kyoto for free with the Rail Pass while you are in Kobe, or do day trips from Tokyo to Nikko in addition to using JR lines around the city. You can also use the Rail Pass to get to and from the airport in Tokyo if it is within the validity period.
For the actual travelling, I tend to just turn up and board the next available Hikari bullet train service. If I have time I also make a reservation when I arrive at the station, but in general have never had any need to reserve very far in advance unless I was travelling during rush hours (7-8am, 6-7pm).
Hope this helps!