Does fasting one day and breaking fast one day in winter, and abandoning that in summer, count as a regularly recurring act of worship every winter, or must the fasting be regular and continuous to include both summer and winter?

Fasting one day and breaking fast the next is the fast of Dawud (peace be upon him), and it is considered the best voluntary fast, as stated in the prophetic tradition. It is not a condition to fast continuously in this manner. For it to be considered the best, it is a condition that it does not weaken the body from fulfilling the rights of Allah Almighty and the obligatory rights of people. If it weakens the body, then abandoning it is better.