The Donkey Kong painting in BK's house isn't out of place or special, as it's not strange to find references to Rare games in other Rare games. They probably removed it by the end of development purely I think just as an aesthetic decision—there's more to it than that though.
In later pre-builds of the game, that DK painting was moved to the one above the fireplace, where Bottles' painting is in the final game and where the secret puzzle minigame takes place. In the final game, Kazooie also calls Bottles "barrel boy" indicating that Donkey Kong was originally the one hosting this minigame. There's no indication that it was ever related to SNS, though.
The BK fridge in Donkey Kong's house would appear to be innocuous, but there's a way to
glitch the game into showing you an unused cutscene. Presumably it would have shown the BK fridge opening up inside DK's house, after the player had just obtained the Ice Key inside Crystal Caves (There is unused text for ICE KEY in the DK 64 pause menu). You would then walk back to DK's house and ???????? special thing would happen.
I forget who said it or which interview it was, but one of the Rare staff members confirmed that Stop 'n' Swop was supposed to be a feature where collecting items in one game would trigger something special in other Rare games—not just from BK to Tooie. They said they only really worked on it for DK64 and BK, as the SNS idea was never fully fleshed out and the idea was scrapped. He said that they basically didn't even KNOW what they were going to use SNS for, or what would the mystery items would ACTUALLY do, but Dragon Kazooie was one of them.