Since Meysan Lake is seldom used for camping, I would highly doubt you would have any bear encounters. The bears congregate around Whitney Portal because of the campground, and the possibility of stealing dayhikers packs in the early morning hours when they are getting ready to hike (they set their daypack down, and the thief makes a grab-and-go run). I've not seen any reports of bear encounters on the Main Trail above Whitney Portal, so the chances at Meysan are miniscule. The biggest problem on the Whitney Trail are the marmots and mice getting into hikers' food. So use a bear proof container, and you will be just fine.
As for rain, watch the weather forecast just before you go. If I were going, and no storms are forecast, I'd leave my tent in the car. On the other hand, any chance of thunderstorms, take the appropriate gear. You can find good weather links in the
Mt Whitney Weather page.
This being a low-snow year (about 60% of normal), you will be lucky to find significant patches of snow in late July.
