Welcome to our Forum. I'll try to help but the ultimate guidance as to what might be going on with your GF will come from the tests her doctor did on her.
Please realize that as an STD specialist, these are the things I think of first and will outline for you with most authority are STDs but if she has an STD it represent bad luck on your part rather than anything else. On the other hand, there are MANY non-STD related problems that could be causing your GF's difficulties and which I am sure that her doctor is checking on.
As far as STDs are concerned the findings of bleeding between periods, an inflamed cervix and pain on intercourse may be associated with, among other things, chlamydial infection. I would certainly consider testing for chlamydia based on the history you have provided. In addition, I would add than men can have chlamydial infection without any symptoms and then transmit it to partners who become symptomatic.
I doubt that your penile rash is an STD of any sort and it does sound like it could be psoriasis or eczematous dermatitis.
My suggestions. You are clearly worried that you might have given your GF something. My advice would be go and get checked, including getting tested for chlamydial infection. While there is a good chance that you do not have an STD and that her problems are not related to an STD or anything that you gave to her, it is important for you to sort this out and reassure yourself. Her doctor will determine what her problem is.
I hope these comments provide some direction. EWH