I have a permit for 13th July and would appreciate if someone can give me an insight of the amount of snow on the trail. Is it really bad currently?
No, there's a lot less snow than normal for this time of the year:https://cdec.water.ca.gov/snowapp/sweq.action
That number will go up a bit after the storm early in the week, but the amount of snow will still be far below average since it's not expected to be a strong storm:https://www.mountain-forecast.com/peaks/Mount-Whitney/forecasts/4418
What do you expect by mid-July? This will be my first time and would help me a lot to plan if I have this information.
No leftover snow, but thunderstorms and hail are likely in the afternoon. Generally, winter storm season is from October - May, and moonsoon season is from July - August. In a light snow year (2014, 2015, 2018, and 2020), you'd want to hike/climb in mid June - early July, after the snow melts and before the monsoon hits. Another good option would be climbing in September after the monsoon but before the first significant winter storms move in.
In a heavy snow year like 2019, the monsoon is usually not as strong or as early as it is in a light snow year, but you'd have to wait longer until the winter snowpack melts. So your climbing window would shift to July instead of June. Of course, the alternate option of climbing in September would work as well.