You are correct in both of your statements.
In 1), "winter" followed by "of" has "the" before winter. This is true not only to describe years, but with other phrases, too:
the winter of our discontent (also a title of a book by Steinbeck)
the winter of the big blizzard
the winter of grandfather's death
In 2), the sentence is correct both with the definite article and without it.
Goggle displays 1,410,000 examples of "in winter," and 1,220,000 examples of "in the winter." Some of the examples in both categories show "winter" as a noun adjunct, though, and many examples of "in the winter" show the phrase in part of a title, such as "The Winter of our Discontent."
"To talk about the seasons in general, we can say "spring" or "the spring," "summer" of "the summer," etc. There is little difference. "The" is always used in "in the fall" (US).
Rome is lovely in (the) spring.
I like (the) winter best.
When we are talking about particular springs, summers, etc., we are more than likely to use "the."
I worked very hard in the summer that year."
A very similar question and answer about "summer" and "the summer" appears on the Newsgroup on page 5. The question was asked by Hogel and answered on February 18 by Rachel.
*Practical English Usage, 2nd Edition, by Michael Swan. Oxford University Press. 1995