Of the examples you listed, the sentences using “…and I” incorrectly are so common in casual speech that I barely notice them. But the examples that start "Me and … " really grate! Reminds me of Cookie Monster. Me want cookies.
In any case, I’m sure it’s a dialect thing for the characters and not any grammatical mistake on the part of David Nicholls.
There is one thing about Nicholls’ writing that became a little distracting after a while: I’ve never read so many similes! Fortunately, most of them land well (such as @Marilyn’s quilt example). But really, there is quite a barrage.
“He was neat and trim, and easy access to moisturizers had given him extraordinary skin, a smoothness untouched by time and life, as if he were his own action hero.”
“Tentatively, they lowered their hoods, like astronauts removing their helmets.”
“The pool occupied a flimsy half cylinder, like a polytunnel used to grow lettuce in winter.”
“It was difficult, too, to sulk off when covered with foam, but she did her best, dripping dirty suds, like a poorly rinsed roasting tin.”
“She showered, put on clothes as if dressing a mannequin, walked downstairs sideways, placing her feet gingerly one step at a time, like a marionette, to the bar, which smelt of beer and fish.” (She gets to be both a mannequin and a marionette in the same sentence.
)
“In these moments, it was as if he were walking within some transparent bubble, like the clear jelly around the black dot in frogspawn.”
Those are just a few of many! Granted, they are often amusing, but also as countless as the stars (dang, it’s contagious).