I was just there last summer and in total have climbed the west side 3 times, descended the west side twice and the east side once.
1. From a road clearing perspective, it is hard to know when the pass will be open. Sometimes hikers and bikcers can get to Logan Pass before cars, but that is not always the case, and the east side of the road might be closed. The best thing you can do is check the park's web site as your trip progresses. The NPS gives daily updates on which portions of the road are open to whom. This summer the entirety of the road did not open until the 3rd week in June due to repair work necesitated by a huge avalanche during the winter. It could be open all the way the first week in June like it was when I was there in '00 or it could be closed into July You never know. We had intended to cross west to east last summer on June 22 but were forced to go around.
2. As noted, there is ongoing construction work that may cause delays and force you to share the road with convoys of dumptrucks. The trucks aren't that big of a deal. When you hear them coming, simply pull over and wait for them to pass.
3. IMO, and assuming you are going west to east, you should camp at Sprague Creek. It's small and thus not a zoo, and there are nice hiker-biker sites for $5/person. Lake McDonald Lodge (a must see) is a short walk away. Check out the room with the trophy heads then grab a beer at the bar and take it out lakeside. There is also a camp store, but buy groceries at the store in West Galcier as the camp store's slection is poor. From Sprague Creek it's about 13 miles to Logan Pass. Start very early to beat traffic, to give yourself enough time along the way to takes breaks and photos and for the best chance at spotting wildlife. If you are there mid to late June, you can hit the road at 6 a.m. and have plenty of light. Starting early is especially important this year as it is the park's centenial so heavier than normal crowds are expected. The first 6 or so miles you won't even know you are gaining altitude. You might see a bear cross the road along this stretch. The next 7 miles vary in grade, but there is nothing too steep. If you keep a moedrate pace you should have no trouble reaching Logan Pass by 11 a.m. When you get there, do the relatively short hike to Hidden Lake if the path isn't too snow covered.
4. If you want to give yourself more time, camp at Avalanche Campground east of Sprague Creek. But note that there is no food there so you will have to carry. Also, because both are east of the Apgar Turnoff mentioned below, you cannot ride to them between 11 a.m. and 4 p.m. If you stay in the Whitefish area, sleep in, ride to West Glacier in the afternoon (heed the map warning about staying off U.S. 2), pick up groceries and head to either campground one the bike ban lifts at 4 p.m.
5. If Logan Pass won't open in time to fit into your schedule, you will have to go around via U.S. 2 from W. Glaicer to East Glacier, MT 49 to Kiowa and U.S 89 N to St. Mary. We did that last summer in the opposite direction. It's roughly 90 miles. You could break it up into two days by staying in E. Glacier, where there are motel, camping and a hostel.