MIKE MOSS SAYS: Tina, Once a thunderstorm produces a tornado, it becomes severe by default whether it satisfies the other criteria for severe storms (hail of .75 inch or greater diameter and/or wind gusts reaching 58 mph or greater), so in that sense the answer to your first question is no. In another sense, and this may be what you were getting at, it is possible for a storm to be less than severe up until it produces a tornado, if the storm suddenly intensifies enough to do so, or in some more esoteric circumstances the storm might produce brief, weak tornadoes of two types sometimes referred to as "gustnadoes" and "landspouts." It's conceivable that these kinds of tornadoes might occasionally form without exceeding the 58 mph threshold. Then, you would have a non-severe storm with a tornado - however, this is pretty much meteorological hair-splitting and stretching things a bit, and it's pretty sensible to just consider any storm that produces hail and wind gusts meeting the criteria above, or producing a tornado of any kind, to be severe.
Likewise with the second question, almost all tornadoes rquire a thunderstorm to form them, but there is room for complexity in that some landspouts and many waterspouts are weak forms of tornadoes that can sometimes form in the absence of lightning. They still require a convective cloud with fairly strong updrafts, but if you wanted to be strict about terminology, they would be forming in the absence of a thunderstorm, since without lightning there is no thunder.
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