This is a good time to see game in the park however, not all of the roads have been cleared of snow and very few of the facilities are open. The highway from Fishing Bridge to Old Faithful isn't open nor is the highway from Canyon Village north to the Lamar Junction. I would suggest you find a website for the park to check the status of the highways before comming this early. I am looking at a snow storm on the mountains around our valley as I write this. It's possible to get a snow squall in July in the park. I would suggest that you plan on comming the last week of May at the earliest and/or the second week of Sept. You miss most of the heavy traffic at those times and find the highways open with reasonably good riding temperatures.
I make several day rides into the park and back home during the season and don't stay overnight in the park but, I would recommend staying at Mammoth Hotel, which is at the north end of the Park. This is why: From there you can ride the loops in the park and also ride east to Cook City, Montana (just out of the NE Park entrance) and ride on east over Beartooth Pass to Red Lodge, MT. Then you can come back south over Beartooth through Sunlight Basin, WY and over Dead Indian Pass down to Cody, Wyoming. In my opinion this is some of the most spactacular scenery anywhere. Then ride west to the Park again through the East Entrance and back north to Mammoth. You cannot ride all of these places in a single day and I would recommend no less than three. Jackson, WY, at the south end of Teton National Park, is a favorite tourist spot and shouldn't be missed. Make your ride through there either comming to Yellowstone or on your way back home. The Teton Mountain Range is quite a site as well and lies between Jackson and Yellowstone.
And as you mentioned, one must be careful of the wild game in and around the park. Especially grizzly and bison. You will be lucky to spot a griz. but bison are plentiful and are often on the highway. Give them plenty of room. They appear slow and docile but can be quick when provolked or scared. And don't ride fast through the Park or mountains around it because of the chance of hitting a game animal. I'll bet the guy who butt heads with the big horn feels like an authority on that particular species now. The Park Service claims about 100 game animals are hit by vehicles in the park every year. Outside of the park watch out for roaming cattle in open range areas. There are also a few human fatilities in the park each year for various reasons; two guys fell to their death last summer off a high bridge near Mammoth. There have been fatalities due to falling in hot springs. And from auto accidents. No fatality but a few years back I saw an elderly couple sitting in their sedan out in the middle of a stream. They were unhurt and no doubt waiting for someone to rescue them, and must have been embarrassed as they were looking straight ahead, not at passerbys.
Let me know if I can help further.