Welcome to Flyer Talk!
Denali NP is my favorite place in all of AK. So much so that I started working near there the past two summers. You say "early June" then you say June 7-10 at Teklanika. So I assume you would be staying in Healy before then. Just so you know, the road is only open to Mile 66 (Eielson VC) from June 1 to June 7.
You really have to take the shuttle from the Wilderness Access Center (WAC). More parking and that is the place to load except Savage Campground and Tek Campground.
Why are you not considering just camping more than 3 nights at Teklanika? The Tek Pass is valid for the entire length of your stay so if you stayed at Tek more nights, there would be no need to buy separate shuttle bus tickets on the earlier days. Sure it means bringing more food with you, but bring a cooler and some non-perishable food and you will do fine. I have been in the park up to about a week at a time at Wonder Lake and/or the backcountry.
Re: car rental. Just a normal basic car is fine. You won't hit snow on the roads you are driving on (there might be snow along the mountain passes in Denali NP but it melts quickly in the morning). I always book the least expensive car rental. Book your car rental NOW. The prices tend to go up the longer you wait. Check off-airport rental offices too since they have far less in taxes and fees vs. on-airport options.
When I traveled to AK with my camping gear in my suitcase, I would bring a tent, sleeping bag (for June a basic one is fine - it won't get below freezing at night), a ground mat (that is important), a small camp stove, a pot and a pan. You have to buy fuel once you arrive since you can't bring it on the airplane. For day hiking, food in your day pack is fine. But in the campgrounds, all food must be stored either in your vehicle or in the bear-proof food lockers provided in the campground.
Since you are camping, you probably don't mind shared bathroom facilities. You might consider the Denali Mountain Morning Hostel for your nights before/after camping in the park. Very well run hostel, clean and a variety of lodging options. They have private rooms, private cabins and shared bunk rooms. I have stayed in most the above options. None have their own bathroom facilities but the showers and toilets are "home style" meaning you are the only one in there and lock the door behind you. They have a nice community area for cooking, eating, and hanging out.
With the amount of time you are spending in Denali, you might consider buying Ike Waits' book on hiking and camping in Denali NP. It gives some good info on potential hikes in the park. You can also sign up for ranger-led Discovery Hikes. They usually have two each day. You can only sign up max 2 days in advance and there are max 11 people plus the ranger on the hike. I have done quite a few of them. One time I was the only hiker who showed up!
Re: mosquito nets. They would be helpful if you are visiting Wonder Lake. Through late July the mosquitoes can be horrendous there. But surprisingly, they are not bad at all in other areas of the park. The only year I recall mosquitoes being bad even near the entrance was 2013. But that was unusual.
If you are planning on hiking, most likely you don't want to take the shuttle all the way to Wonder Lake. That is a long day and would extremely limit your hiking options. The other option is to camp at Wonder Lake (my favorite campground in the park) and then go from there into the heart of the park each day to hike. You would have to battle the mosquitoes, but a headnet and a mosquito lamp helped me greatly. Then you take one of the two early camper buses in the morning back into the heart of the park to do your day hiking. The benefit of this option is you have only 28 campsites, and no one drives - they have to take the bus to get there. Plus, when you are heading from Wonder Lake eastbound into the heart of the park, very few others from the entrance will get to the heart of the hiking area before you do. It is like having the park all to yourself. Then at the end of the day you hop on a bus back to Wonder Lake for dinner and spend the night. The only issue with WL is that you could not camp there until Jun 8 when it opens.