Yes, that is a typical phrase when we close a letter. As a direct translation to English, that means: ”Please take care of yourself and I hope you’ll always be fine.” Even though it’s a common phrase in letters, you can still use daily conversations with anyone. However, about your third wondering, いつも気をつけてね isn’t the same meaning and we seldom say いつも気をつけてね. I guess you might regard　気をつける as “take care”, but when we use 気をつける with no objects such as 体 or 健康, 気をつける almost always means “be careful”.　I’m sure this is a very difficult aspect of Japanese language, but please don’t worry because the listeners will surely understand and appreciate your kindness even if you say so.