You cannot summit Half Dome up the cables with a wilderness permit out of Mammoth. You can if you reverse the route and start at Happy Isles. So that is something to consider if you want to summit Half Dome using the cables. If you want to summit Half Dome and don't want to reverse the route, you can try and snag a stand alone Half Dome permit when they become available (i think you need to submit your application in March or April). Summitting Half Dome up the cables is an unforgettable experience, so I would definitely consider it.

I recommend going in August if possible. June will probably still have lots of snow over Donahue Pass. Late June to mid-July (or late July on a year with a healthy snowpack) can have massive swarms of mosquitos). August is ideal.

This is a very popular hike because it includes both the JMT and PCT so I would go on recreation.gov and familiarize yourself with the permit process. I would request permits the first day they become available. There are a couple different trailheads that will get you to Thousand Island Lake out of Mammoth. I recommend the River Trail (JMT) or the High Trail (PCT), but you can also get there using the more easily obtained Rush Creek trailhead if the other two options are unavailable.

Lots of bears at Thousand Island Lake. Don't leave food unattended anywhere in the Sierra, but be super careful at Thousand Island Lake.

You are going to have about a 2.5 hour shuttle between Mammoth and Happy Isles. There is a bus that runs back and forth between the Valley and Mammoth. Its called YARTs. I would reserve a ticket early if you decide to go that route. Alternatively, you could dump a car at Happy Isles, but that would be a five hour back and forth. I've hitch hiked from yosemite to Mammoth and its actually one of the most common ways to get around between traiheads in the Sierra, but I would choose that as a last resort.