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For question #1, highly unlikely. Contrary to the way it may seem from watching Weather Channel, Buffalo is not above the Arctic Circle nor is it in Siberia. The average daily highs for April in Watkins Glen, Syracuse and Buffalo are all in the 50's. More than warm enough to melt off any snows that lingered into early April.
For question #2, if there does happen to be snow, I wouldn't worry about the roads. Having grown up in Maine and spending 4 years in college in upstate NY, I have found that the areas that get the most snow (like the Buffalo area) tend to do the best job of maintaining safe road conditions in the winter. Over the years, they have learned to manage those conditions better. I now live in NJ and I'm embarrassed at what passes for snow maintenance. I'd feel safer in an economy rental in NY than 4 wheel drive in NJ.
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Early spring weather in May should be beautiful, perfect for visiting the area.
BTW, Letchworth is a NY State Park, not national. Excellent though, as is Watkins Glen, also a NY State Park.
Additional suggestions -
If you are in Buffalo, Niagara Falls is just a little over half an hour away and is very much worth a visit. It has many fabulous viewpoints on both the American and Canadian sides and the border crossing into Canada usually goes pretty smoothly and quickly. Goat Island is excellent. Aside from parking, the best stuff is free.
Also worth a stop is Chimney Bluffs State Park in Wolcott, between Rochester and Syracuse.
Buttermilk Falls State Park in Ithaca and Taughannock Falls State Park north of Ithaca are also worth visiting.
As people have said, don't worry about the weather. Not that you will have snow in May, but just for the record, my sister and brother in law lived near Rochester for years and the kids never missed a day of school due to snow.