First question:
I haven't WiiU, but exactly what's happen if I use WiiShop on Wii after a restore of my NAND (where I had some Wiiware) ?
Normally, when you do a system transfer to your Wii U, all legitimate games that were purchased in the WiiShop are removed from your Wii. They will no longer work on your Wii and you cannot download them again on the Wii. You can only use them on your Wii U.
If you did a NAND backup before doing the transfer, you will restore all of the games that were removed and you can continue to use them. If you visit the WiiShop, these games will be modified so they cannot be played. They will still exist on the Wii, but you cannot use them. You'll have to do another NAND restore to get them back.
All works good, but seems that now, on Wii, some exploit doesn't work.
BannerBomb and LetterBomb stop working (he try to install different System Menu)
But games exploit and Wilbrand works.
Is possible? Why?
The transfer itself does NOT modify the Wii system. However, you are required to update the system menu to do the transfer (because the tool must be downloaded from the WiiShop).
Nintendo patched BannerBomb in system menu 4.3. However, LetterBomb
only works in system menu 4.3. There is no way your friend had both BannerBomb and LetterBomb working at the same time on the same Wii. You mentioned that he tried to install a different system menu but you did not explain how, what versions, which version he tested each exploit on, etc... I can't explain what exactly is going on with your friend.
I can say that as long as you do a backup before the transfer, everything will work exactly as it was after you restore it.