I had a recent exposure with a Brazilian escort. The oral sex was covered, and so was the penile-vaginal portion; however, I felt like my penis slipped out of the condom and had several inserts that were bare inside her vagina; however, when I finished, I saw that my penis and my semen were all inside the condom. Would it be possible that my penis somehow slipped out of the condom, entered her vagina, and then slipped back into the condom? If so, what are the likelihood of gonorrhea, chlamydia, and hiv transmission if she had it?

Also, after the sexual interaction and I was cleaning up. My penile opening made contact with the outside of the condom as I was removing it. Is there a transmission risk there? Thank you

Your exposure was safe, fully condom protected and therefor no risk.  As you no doubt know, condoms are quite soft and flexible.  As a result, when they slip off during sex they immediately are pushed deep down into the vaginal and do not ever get back onto the penis.  You can be confident that your condom did not come off and therefore, you were protected.  The presence of ejaculate in the condom after completion is proof that the condom stayed in place.  

Similarly, the fact that the tip of your penis may have come into contact of the outside of your used condom during the removal process is a no risk event.

From my perspective, there is no medical reason for testing or concern.  Of course, if you are concerned, testing is always a personal decision.  

I would not worry.  EWH