Most if not all HDTVs will play both 50i and 60i high definition video. 25p, 30p, and 24p video is streamed as 50i (in the case of 25p) or 60i (in the case of 30p and 24p) when playing directly from the camcorder, in most cases (exceptions like Canon's HV40, which I believe shoots "true" 24p, are rare in the consumer camcorder world).
As for editing, always match your import settings to the video you are importing. In the OP's case, this means using whatever preset his or her software provides for editing AVCHD 1080i60 video.