What happens to Eureka and Renton at the end of "Eureka seven"?
According to Wikia,


  In the original Japanese version, the final scene of the episode revealed a colored picture of Renton and Eureka with Axel and the children, which confirms that they have returned to Bellforest as promised.




which is actually an end card. However, end cards are usually not considered as canon, so just interpret this picture personally.

As for why the children didn't appear in the sequel, most of the staffs who worked on the sequel are different from the original, including Dai Sato (the chief writer for the original). This thus resulting in many plotholes. Some fans, to some extent, also disregard the sequel as canon due to this.