I'll focus on your final sentence - "I am worried about ice, weather, and odds of getting a permit." Your odds of getting a permit are maximized by applying for dates past Labor Day. Permits for summer dates go to the lucky - we've been shut out for two years in a row for the "traditional" summer days, but we got a permit last year for Sept 6-9th. There are fewer people on the trail once Labor Day is over, and less/no ice and better weather.
Here's my other comment: Yes, there will be no reserved permits available from mid-June to Labor Day once the lottery is run. However, if you fail at the lottery, you will still have a good chance of getting a permit for mid-week days by showing up to the Interagency Center in Lone Pine. A significant # of people have their days reserved and permits paid for well in advance, but cancel their trips or don't bother to show. It's worth the risk, but I understand if you have anxiety because you don't live close by.