You aren't all messed up at all. But, believing that you are is really messing up your head--and I do mean your little head. In other words, worrying about whether there is something wrong with you, is enough to blow your entire show.
Here's why: In order to become fully aroused, a person must be in a relaxed state (this is called the parasympathetic mode). In this parasympathetic mode, blood vessels distend, which allows the penis to become fully engorged with blood. If there is the slightest worry, blood vessels constrict and the blood can't flow fully into the penis, which makes it impossible to achieve a fully'locked' erection. Signs of an incompletely locked erection are a penis that isn't completely hard, and/or difficulty sustaining the erection. Remember, worry alone is sufficient to cause this'plumbing malfunction.'
A man also needs to be relaxed to achieve an orgasm. So, as you can see, the fact that it takes an hour for you to finally orgasm, is fully understandable, given that you aren't relaxed in bed.
Regarding your next question--why you are able to get harder the second time around. This also makes complete sense. After you have finished the first round of intercourse, you have a little confidence under your belt--no pun intended. So, the second time around, you don't worry as much, which means that your body relaxes, and you become harder than the first time.
And, regarding your final question--why you feel like you're having an orgasm, but never ejaculate during your'second inning.' This is also normal. After a man ejaculates, his body needs time to reload--meaning, the body needs a rest period before another ejaculation can occur (called the'refractory period'). The length of the refractory period varies according to many factors--age, health, etc.. Very young men may only require a ten minute rest period between ejaculations. Whereas older men may require an hour, several hours, a day or even several days in order to be capable of ejaculating again.
Note that a man can still achieve an erection even though his body isn't ready to ejaculate again. And, this is what is happening to you. You become hard, have intercourse a second time, and you even have orgasms, but you don't ejaculate, because your body is in the refractory period. So, your girlfriend may not believe that you can orgasm without ejaculation, but, in fact, men can and do, especially during their'refractory periods.'
There is one thing more thing I need to tell you. It feels like you and your girlfriend are scrutizing your penis under a microscope. No wonder you are suffering performance anxiety. In fact, I'm surprised you can get hard or orgasm at all.
I want you and your girlfriend to stop putting your penis on trial. You are absolutely normal and fine. And, if you both stop all this pressuring, you will relax, become harder and even orgasm more quickly. And, when this happens, I don't want to see your girlfriend point the finger at your penis and tell it,'you are coming too fast!'
Relax and enjoy. You are fine.