I think you answered yourself above with the fact that the tires may be slightly stretched. This can lead to wheel damage, of course. Once the wheels are refinished and you have meatier tires on them, you should be good to go.
Poster above mentioned 265 and 305's and that would be fine, as, like he said, plenty of people run that sizing. However, you can get by with smaller than that, if you don't want to go that wide. It's up to you.
As far as tire upgrade, I would not choose the RE11, if I were you. You said rain was a concern of yours and I'm sure the RE11 does fine in rain but if your looking for a grippy summer tire that has good review on wet pavement and a little better longevity, go with the Potenza Pole Position S-04 (I currently have them and love it) or Michelin PSS. The PSS may be a tad better and last a little longer but it may not be significant enough to account for the price difference. S-04 would be just right for you, in my opinion.
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"Let me tell you something. Anyone who owns a car like this (370Z) has something wrong with them in the head." My Grandpa who has owned 4 Z's in his day. 
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