The One With Joey's Porsche - S6-E5
Factual error: Phoebe discovers that one of the babies has wandered away, and finds the baby inside the open drawer. Yeah, no. These babies aren't even walking yet, much less are they able to lift themselves up past the height of their own bodies and launch themselves into a drawer.
Factual error: This mistake is from series 6 onwards, with Ross' new apartment: There is an impossibility with Ross' kitchen, but not all of the time. It depends on the layout of the hallway outside his apartment. If the hallway continues along for a bit before going left or right, then the kitchen should stick out. However, when the hallway is a dead end, the kitchen fits ok.
The One With The Proposal, part 1 - S6-E24
Factual error: There's no charity event in the world that would be clamoring for non-paying guests and offering them free food and an open bar.
The One Where Ross Got High - S6-E9
Factual error: A really picky one: When Ross looks at the recipe book, he says Rachel's made 'half a Shepherd's Pie'. Technically, a Shepherd's Pie is made with lamb, whilst it's a Cottage Pie that's made with beef (although the two names are often interchanged).
Factual error: The makeup on Rachel's face "can't be covered up with makeup," and "won't come off," we learn in The One In Vegas, Part 2. So it's not a surprise that she still wakes up with it on (and so does Ross.) but an hour or so later, when she's at breakfast, it's not there. An hour or two won't make the magical difference that 8+ hours didn't.
Answer: The statements that she's contradicting (that it is common, it happens to every guy, and it's not a big deal), are the things that a woman commonly says to a man who is suffering from erectile difficulty, typically to assuage his bruised ego. However, most men do not believe that these statements are true, as evidenced by Chandler's outburst. He's so caught up in the proof that women are lying about it that he gives himself away.
Rooster of Doom