James, by the end of June, your trail runners will be sufficient.
And whatever gear you have for the entire JMT is all you need for Mt Whitney. If it is stormy, you'll probably choose not to be up there, but you can always spend the night in the hut if you need more shelter than you already have.
Overnight temps depend on the current weather -- you can get really cold conditions, or perfectly pleasant (if cold) conditions. Look at the sky and decide when you are ready to go. Just take enough water from Guitar Lk to last overnight.
