1. I would not take the ice axe. People are doing just fine with microspikes and poles.
2. Diamox: (Too bad you didn't get the 125 mg tablets.) AMS doesn't show up for most people until 12k or higher. First, try it at home: take 1/4 tablet twice, 12 hours apart. If no side effects, good -- follow your plan starting 7/17. ...just to be safe. I follow your plan myself.
3. gps: You will be just fine. It's not so easy when some of the trail is buried. But not mid-July.
4. Acclimatization: what Bob P said. Horseshoe meadow always has available walk-in sites (one-night only), so go there 7/17. Use the Whitney Portal walk-in hiker sites on 7/18.
(Link)If you get to HM by noon, go ahead and do Trail Peak. Then spend the night by your car. Next morning head down to Lone Pine, pick up your permit, head up to Whitney Portal. Get a walk-in site. Then, hike up the first mile of trail, cross North Fork Lone Pine Creek, take
the Old Trail down (or the reverse).
If you do the above, you will likely sail up the trail on 7/19 with no AMS issues.
5. what Bob said about Lone Pine campground. Check with the Portal Store about closing times.
#6: Have a great trip!!
