As your Wii is a 4.0U, there is no compelling need to update to 4.1 as there is little to be gained from that update. Don;t worry about about updating to 4.1. To answer your question, however, WiiConnect24 OFF, Internet ON to connect to the updating server. If WiiConnect24 was left ON your Wii would have connected to the Nintendo server and been updated to 4.3-something you definitely do not want.
Priiloader-sometimes the hack.ini file does not download or extract properly into the root directory of your SD card. That has happend to you. Go HERE to redownload thecorrect Priiloader hacks.ini file and that should allow the hacks to be seen and enabled. A 4.0U is the same as a 4.1U Wii s choose the 4.1U hacks.ini file.
While the NeoGamma version here is a bit older, the only difference with the more current version is not one that has to do with Wii games, but address a GameCube issue. I suggest you reinstall NeoGamma and try again or just check the NeoGamma settings. A certain amount of trial and error can be needed to play some out-of-region games. Try different combinations of Force Video, Patch Video, and VidTV. For your NTSC Wii, set the Force Video to either Force NTSC or to Console-that works most of the time.
Best wishes to you and your twins. Hope they have a happy Christmas.