Harry's accent gets thicker and thicker as he's succumbing to the Truth Serum, and the way he constructs his sentences gradually begins to sound more like German than English.
Remember Dana in the beginning of the movie managing to steal Harry's partner's money from his coat without them knowing? It comes in handy when she steals the key to a nuclear warhead while being hostage to a room full of terrorists without being noticed.
Considering this movie's Reality Subtext, you can easily read the plot as a Take That! from Cameron in regards to his marital troubles: "You want excitement in your life, honey? Ha! Well, how's this for excitement? You think I'm married to the job? Well, at least I'm not using my job to be a creepy stalker!"
Harry and Helen get captured by the terrorists because Harry grabbed the Idiot Ball. Of course, if Harry hadn't been captured, he never would have found the terrorists and the nukes in time. Also, the whole experience helped reinvigorate his marriage and improve his relationship with his daughter.
Gib's Brutal Honesty regarding Helen's affair may seem insensitive—"What'd you expect, Harry. Helen's a flesh and blood woman and you're never there"— until you recall that Gib is the veteran of several failed marriages himself, which were very likely due to him never being around.
Whatever elite spy organization Harry works for seems to be amazingly tolerant of their agents using a lot of money, time and resources to stalk their loved ones. Harry's partner does point out repeatedly that he shouldn't be using agency resources for his personal issues, but Harry doesn't listen. When his partner flat out tells Harry he cannot call on-duty surveillance teams to follow his wife, Harry reveals he knows his partner screwed up a big operation because he was getting sex from a hooker, and kept quiet about it. This results in the partner having to cover for Harry with the never-seen Big Bosses a lot. It seems a lot of other agents, contacts and assets owe Harry favors (or he has leverage over them) that he can call in at his discretion. Which is actually not unlike real world spy networks.
Also, Harry and his wife, presumably, sometimes have sex, which presumably involves both of them being naked together. It stretches credulity to believe (especially given the nature of Harry's assignments in the film) that he has never picked up any scars or injuries on his body that she might notice and wonder what her husband has been doing to get them.
Harry's a convincing and charismatic liar. Considering his cover, he probably made up an extremely long, extremely detailed, extremely boring explanation for how he picked up the scar by falling on the luggage turnstile at the airport, to the point that Helen would have probably tuned out and just figured that even Harry's hurting himself was dull.
How did the terrorists get Dana from Washington D.C. to Miami in under 90 minutes? We know it has to be less time than that, because they don't find her photo until after the bomb is set to that time limit. Even if they had members of their cell still active in D.C. to grab her and bring her down, you would need a plane capable of MACH 1.72 to make the trip that fast. No civilian aircraft could do that. And Middle Eastern terrorists certainly weren't going to get their hands on US military aircraft. Did they steal the SR-71 from the Smithsonian?