Will be in GNP July 14-19.

First- Is there any word at all on when the Going to the Sun Road will open?

Should I consider packing all long sleeve shirts? We plan on packing plenty of items to layer, but then it struck me that we might actually get hot and might be way over packing. It is hard to tell what we will hike, because it seems that a lot of the trails might be closed due to the large snow this year.

Throw in at least 1 T-shirt and insect repellent. When the snow does melt, the mosquitoes and other bugs will be out for blood.

You might indeed get warm. Layers are good with a t-shirt on the bottom.

Layers is the key.
GTTS opening is undetermined at this point. Plows have pionered a path from the west side to Logan Pass. Plows on the east side are surveying the Big Drift to find the road. Educated guess is week of the 10th of July. Road is open to Avalanche Creek with free shuttles able to take visitors as far as the loop.

I have my fingers crossed that it will be open by the time we are there.

Long sleeve shirts are a good idea owing to the intensity of the sun at high altitudes, cold nights, and bugs. But you will want lighter layers as well. A hat, preferably more than a baseball hat, is also a good idea.

It could be really warm during the day and really chilly at night. You will need a little of everything. A light long sleeve shirt and pants are great if you are sitting around outside to keep mosquitoes off of you. They don't just come out at night either if they are really bad. A lot of times there will be a good breeze or wind--sometimes more than you want--that will keep them off.

Temps. here in the high 70's all this week which should help with the snow melt. I suspect the road will be open by the time you get here. Rivers are RISING rapidly though. Avalanche Creek trail and Apgar Lookout trail closed due to flooded trails.
Granite Park chalet open with access from the loop only . Traffic not allowed past Avalanche Creek EXCEPT for shuttle buses which are running from Apgar Transit center to the loop.

Now the question. What to hike? And what will the trails be like. I had originally planned on hiking The Highline trail as I normally hike "the spectacular" at each National Park I visit. I also thought about Iceberg Lake. However, my wife and daughter say they wanted to hike easier hikes. I thought about the Hidden Lake Trail, Trail of the Cedars, Avalanche Lake. Will Hidden Lake still have so much snow that it will be impossible?

We have lake cruises for about every day. Have a Red Bus/Tea at Waterton planned as well. Also plan on doing the zipline and alpine slide at the resort.

Pretty sure the Highline won't be opened. They need to get Logan pass open first then get the safety lines attached then clear the trail. Set to start clearing July 15.
Iceberg lake is also iffy. Clearing from Ptarmigan cutoff has started.
Hidden Lake overlook will most likely be passable but snow covered in most places.
Avalanche, John's loop, Rocky Point all pretty easy and all opened at this time

GTTS road opens officially July 13. Highline trail is still heavy snow pack at this time as are most alpine trails. Best to check at the visitor's center upon arrival as coniditions change quickly this time of year.

Which resort has the zipline & alpine slide?

Big Mountain-http://skiwhitefish.com/activities_zipline_tours.php
Sounds like a thrilling place.