Not for the city itself. Downtown, the waterfront, the Hill Section and UVM are all walkable, and Green Mountain Transit buses cover the rest, including the airport three miles east. You will want a car for Stowe, Shelburne, the Champlain Islands or any hiking.
September and early October give the best weather and the foliage, but they are also the busiest and priciest. July and August are for the lake. Winter is cold and often grey, with the cheapest rooms; April is mud season and worth skipping.
Yes. North Beach, Oakledge and Red Rocks Park in South Burlington are the usual spots, and the water is comfortable from late June through August. Vermont posts occasional cyanobacteria advisories in late summer, so check the beach signage before you get in.
The Ben and Jerry's factory in Waterbury is about 35 minutes east on I-89, and Stowe roughly 50 minutes. Both make easy half-day trips, and you can string them together in one loop.
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