I had a 2007 FX-35 a few years ago that had the hard starting problem. My story may be useful for others.
I bought the car used in 2010, and almost immediately I started having the hard start problem, mostly in cold or wet weather. My dealer was hopeless. They replaced the fuel pump under warranty, which didn't help. After a few more useless attempts at diagnosing the problem, they gambled on replacing the battery (which had tested fine). This worked, for a few months.
When it started happening again, I didn't bother with the dealer. I bought another battery with the highest CCA I could fit in there. This helped, but only briefly. Then I tried an expensive Optima battery, which helped for a bit longer.
Next, I got on Google and started researching the problem. I came across a single obscure post that said replacing my 27760-8Y70A AC Amplifier module (which I still have) with the newer 27760-8Y71A might help. It made no sense to me, but I didn't know what else to do, so I tried it. It worked! After this, I had no problems for maybe 3 years.
However, after 3 years it burned out a coil. My wife kept driving it for a few kilometers with the Check Engine light flashing, killing one of the catalytic converters. Shortly after replacing the coil and converter, another coil burned out, and I could see by the bulge on yet another coil that it would soon burn out as well. I pulled out all the spark plugs (one of the first things I'd replaced in 2010), and the gaps had gotten huge! This is very unusual in modern cars. I decided the coils were burning themselves out attempting to create enough voltage to jump those large gaps. After replacing all the spark plugs again, I had a few weeks of peace, and then another coil burned out.
After replacing this coil, and continuing with my theory that the coils were burning out from having to work too hard (even with new spark plugs), I decided to ensure the entire path back to the negative terminal of the battery needed to be cleaned up as much as possible, in order to minimize resistance. I took off every bolt and fitting connecting the engine to the ground cable, to the body, then back to the battery. I sanded and cleaned them so that bare metal was touching bare metal all along the way, with no rust, grease, or even paint to increase the resistance. After this, I had no more trouble.
After cleaning up the ground path seemed to have fixed my problems, I wondered if the new AC Amplifier never was the problem, but somehow the replacement one allowed the ignition system to run a higher voltage, which fairly effectively covered up the grounding problem, at least for a few years (but ended up killing a few coils). I never did test my theory, though. Unfortunately, the constant reliability problems had resulted in my wife, the principal driver, having a special hatred and distrust of the car. A year later, we sold it. I really do think I'd finally fixed its problems; I hope so, for the sake of the next owner.
........ no idea.