If you're going to enjoy it, you ought to just go with the narrative flow--more emotional than logical. There's a cameo for Stefania Sandrelli as the red haired hitchhiker; really there's no point to the anecdote except to give Vanel a chance to throw the money away, and Belmondo and the girl to pick it up. The Sinatra detour is just time wasting but it's also atmospheric. The stripper is most unlikely, but probably Melville wanted to give a part to the actress (was she his mistress at the time? who knows).
It's a better film than Two Men in Manhattan (not sure of English title) which was really clumsy, and had worse acting.
I much prefer Two Men in Manhattan. To me, that's a really good movie. This doesn't even flow well on an emotional level. It looks pretty, but not much else. Some individual parts, like the sinatra detour you mentioned worked for me, but nothing meshes.