Bean's solution to the destruction of the Whistler's Mother painting in the movie is never gonna hold. As soon as somebody gives it a closer inspection, or (accidentally) touches it, or it's up for renovation of whatever sort, or is ever moved, the switch will become apparent. If the movie had a "Where Are They Now?" Epilogue, both Bean and David would have been discredited professionally and incarcerated for being complicit in a conspiracy to destroy and steal a national treasure said to be worth fifty million dollars, and David would also lose his family forever. Bean just might be able to get out from under it or get a reduced sentence with an effective defense arguing that he has diminished mental capacity (putting the blame with the board of the British National Gallery instead for nominating him in the first place), but David's life is absolutely and irrevocably ruined. Which, to be frank, is partly his own fault. When Bean presented to him what he had done to Whistler's Mother, David should have immediately reported this to his bosses, not try to cover up for Bean.
On the other hand, there's always the possibility that they could blame the French government, assuming they sold them a forgery in the first place? Or assume that it had been stolen before it had even arrived at the Gallery?
The painting was put in a sealed vault that was not going to be opened for a long time. So no checking or renovations to the painting for a good while. Though it's still an obviously flawed plan.
Yes, because that just delays the moment until they'll be caught. Maybe they manage to get away with it for several years, but when the museum inevitably finds out that they have a fake, all they have to do is recheck the video evidence of Bean destroying and replacing it. (If David hasn't deleted those tapes already.)
Deleting the tapes is kinda a given. They show not only Bean destroying the painting, but David participating in the concealment of it (also maybe they record audio too which would indicate David's full awareness of the consequences), and finally the cover-up/replacement. Those tapes would probably be checked very soon after they were recorded (and then put into storage to cycle in new tapes), so David may have even gone to the point of immediately deleting the first tapes showing up to the point of his involvement before he took Bean on the binge.
The script book has an unfilmed ending in which a slight earthquake damages the picture's frame. Someone walks past it and notices the words: "THE GRIERSON GALLERY, HOME OF WHISTLER'S MOTHER." Cue Oh, Crap! moment.
People would still have no obvious reason to believe that Bean and David were the parties responsible depending on how long it is before the truth is uncovered, particularly if Bean's fake reputation still stands up to scrutiny.
They would have no reason to really believe either were involved, being that both David and Bean were never around the painting long enough (to their knowledge) to actually steal/replace it, they would probably investigate the people involved in the paintings transport from France to America. Basically David and Bean are on the bottom of the list of potential art thieves.
Early on, Grierson dismisses the meeting because he has to go to a meeting with the mayor. In his own words, "Not something I enjoy doing, but duty calls!" An apt description of David's "coming to understanding" of Bean, and also Bean's speech about the painting. They both come to enjoy the tasks by the end, though.
Why is Bean able to distract and disable the museum security guard so easily? Because that's essentially what his actual job is, and considering his Cloud Cuckoo Lander tendencies and terrible job performance so far, chances are that all the things he subjects the poor guy to have probably happened to him at some point during his shifts.
