Hello! Any trip to Disney can really be a blast, even if it feels like a shorter trip. If it were me, this is how I would plan my days:
June 28th - Arrive at Animal Kingdom Lodge about 1pm, head to DHS until close at 10pm.
June 29th - Epcot all day (9am-9pm), Easily hop to DHS (walk or boat or bus ride of 20 mins or less) if interested (EMH until 12am).
June 30th - MK all day (9am-midnight).
July 1st - 8am EMH at MK to hit anything you missed, then AK by noon until close at 7; Or, AK all day (closes at 7pm) and then hop to MK (closes at midnight).
July 2nd - AK 8am EMH for anything you missed; 10am checkout and go to Universal.
In my experience, EMH are awesome for that hour or two that you get in the park before or after the masses are there. However, those extra hours draw thousands and thousands of resort guests to that park who stay for the rest of the day, making the park extra crowded. We try to strategically use EMH as needed, and if not utilizing then we try to avoid the EMH park for that day because of the extra crowds. (Also, when Epcot has morning EMH from 8-9, the World Showcase still doesn't open until 11am, which is a large part of that park.)
MK is the busiest park, and it is usually the worst over the weekends and on Mondays, so I would try to make it there on Tuesday or Wednesday.
Note that DHS has gained a reputation as a "half day park," which is why I would go there the afternoon of your first day.
Park hopping does help for days when certain park hours are shorter or longer and you finish up in a park. However, park hopping does always wear me out. I enjoy my days more when I can spend a full day focusing on one park and not having to worry about busses and monorails and boats and travel time and cramming as much in as possible. I think the need for park hopper depends on your type of Disney vacation - if you want to get every ounce of Disney into your time as possible, spring for the park hopper. If you are a more leisurely visitor who goes at a slower pace, you can just focus on enjoying one park at a time. Make sure you get to the park as soon as it opens in the morning and try to hit whichever must-do items you don't already have fast passes for later in the day.
You can do the parks in four days. Know that there is a lot to see and do, so Identify which rides or shows are most important to you and your group. Set up your fast passes as soon as you are able to in order to guarantee you get to do the options you want most. Try to have a plan for the day and know what things you really want to do.
And then just try to have fun!
