The Japanese kanji used to write gyoza are 餃子.
These two characters are also used for Chinese jiaozi, although their pronunciations differ between the languages. In Japanese, 餃子 is read ぎょうざ, or gyōza. The first character, 餃, has the food radical on its left, while 子 means “child” in many contexts and also appears in the established name of the dish.
Gyoza is a Japanese variety of jiaozi rather than an entirely unrelated food. Japanese cooks adapted the dumpling after recipes and culinary knowledge arrived from China, particularly through people returning to Japan after World War II. The Japanese version is commonly pan-fried and often has a strong garlic flavor, while Chinese jiaozi vary much more by region and may be boiled, steamed, or fried.
The alternative kanji in the question are different food terms: 春巻 means spring roll, 饅頭 refers to a steamed bun or related dumpling term, and 焼売 is shumai. They are not alternate spellings of gyoza. Kana such as ぎょうざ show the Japanese reading; 餃子 is the standard kanji representation.