Hyundai Repair: What do you think it could be?, fuel pressure regulator, purge valve

Question
We have a 2001 hyundai sonata with a 2.4 L engine. It has 3 seperate problems I believe.

The first problem happened a little bit after we just bought this car. When we bought it was somewhat warm and didn't notice any problems but as has gotten colder It is becoming
harder to start. It will crank normally no loss of power,no slow cranking just takes a while to build up fuel pressure. however will take several seconds to start between 3 to 15 seconds depending on temp outside. Sometimes have to put the foot on the gas pedal to get it to start I know with fuel injected vechicles you should not need to. It doesn't have to sit for long periods of time for this to occur, only
sitting a half hours time we have encountered this. Once it is fires up and idles normally no problems at all. Drives fine Idles fine. I have read over a lot of previous questions and It seems to me it is either my purge valve although less likely or the fuel pressure regulator. What are your thoughts as far as what the problem can be. (This was intially my first problem it doesn't idle normal at all due to misfire, I believe this original problem is caused by a faulty fuel pressure regulator how do I verify it is bad).

Problem 2 started when we were on a long road trip from omaha to kansas city we pulled on the side of the road to get something to eat when we got back in the car the engine was surging much higher to get going than normal. As I got on the interstate the rpms were running much higher than normal and there was a hard shift like it slammed into gear at about 40 mph. At 60mph it was running at 3500 rmps compared to normally being a little over 2000 rpms, It sounded like the engine was racing hi but it just wouldn't shift into the next gear. I got home and put in on code reader and came up with p0715 and everything I have seen associated with this code by researching tells me it is the transmission input speed sensor I am basically just looking for clarification of this as well. It would come and go for awhile but now it seems more constant.

My 3rd problem was that I had an engine miss, I pulled my plugs and replaced them and noticed that plug number one was fuel fouled was wet with fuel as others were dry, I replaced them on saturday well today while on the interstate the miss came back. I got the car back home going barely 60 on the interstate and slower up hills. I read the code miss fire cylinder number one again. I pulled it and this time it was fouled with a black powdery substance, I replaced it and it improved little if at all. I check spark to the plugs all had good spark. I am pretty sure it is my injector on the number one cylinder that is bad but looking for confirmation before I change it. I will ohm the injectors, and test the harness with my noid light later tonight as well as run a compression and fuel pressure check. I was just wondering if it sounds like the fuel injector to you as well.

Thanks for answering my question.  

Answer
Problem 1:  Your assessment here is fairly sound.  To check the fuel pressure regulator, you'll need to install a fuel pressure gauge, start the vehicle, and then turn it back off.  Normal fuel pressure is 38-50 PSI while running.  So I'd expect the regulator to hold at least 15-20 PSI while the car is off.  

Problem 2:  Everything you list here is consistent with the input speed sensor being the issue.  I've never seen P0715 occur when the issue was with anything other than the input speed sensor itself.

Problem 3:  Be sure to check the secondary ignition system well prior to moving on to the fuel system.  Nearly all misfires are caused by the secondary ignition components.  If the cylinder in question has a wire on it, pull the wire off the plug and place back on without clipping in place.  Then gradually pull the wire off the plug and observe the spark.  If it starts coming out the side (rather than the bottom) of the piece that goes down the plug tube, then you have an issue with the wire.  Also check for arcing from anywhere else other than out the end of the wire, such as from the coil.  Also inspect the plug for a carbon track on the ceramic.  If you suspect that a coil is the issue, you could swap coils and observe whether the misfire moves accordingly.