Talk to your urologist. If you don't have one, ask your regular doctor for a recommendation.
Hey man, it sounds like to me that your penis from the base, the entire shaft and head, and all just points upward, parallel or basically parallel with your torso, but that there is no actual bend mid way or anywhere along the penis. Did I read your post correctly? Peyronie's is when the penis itself is not straight. Your penis sounds perfectly straight but it just does not point out away from your body, it points up towards your face, right?
If I read that correctly then you are describing the healthiest a penis can be! PLEASE do not mess with that. If you mess with this, you'll regret it. If you've seen porn etc, yes a lot of guys in porn have their penis pointing straight out. There is a risky surgery to do that, so perhaps they've done that or their penises are just not as healthy as yours.
Also when you describe it being rock hard, with no flexibility, that is also the healthiest a penis can be! Any softness at all while erect is a sign of a problem. You do not want the penis to get "a little" softer (the girl does not want that either). That would mean you are damaging it. So do not try to do that. A softer penis also will mean a smaller penis. The way the hardness works is it's about tissue expansion. When the penile tissue is fully expanded, it is trapping the blood inside the most, keeping it the fullest, thus hardest. If you cause some damage, creating "venous leak" for example, the penis will be softer, but also this means less fully expanded, so smaller.
Blood rushes into your penis when your body gets the nerve signals to do so, and this blood causes the tissue in your penis to expand. The tissue expands pinching the veins in your penis shut. Veins take blood out of your penis and back to your heart, so you want the veins closed so the blood stays inside your penis, so you can be rock hard and stay rock hard for as long as you need. Then when you are done, and arousal goes away, then the tissue starts to retract, the veins open up again, and then the blood can leave the penis. If the tissue gets internal scarring, for example, it will not be able to expand as well, and will leave the veins open to some extent, so some blood will exit the penis, making it less hard, and less able to maintain an erection (Erectile Disfunction). Too much damage, and the tissue will expand so little, that as soon as blood rushes in, it rushes right back out, and people don't get an erection at all etc. So softness is a sign of a problem.
You can still have sex with your penis the way it is, you just have to be fairly parallel to the girl, so basically be over top of her like that. If you want her over top of you, please be careful, but she'd have to go with your anatomy. You may not be able to stand up with her laying down.
Be careful that the girl knows not to pull it so far down, as that can cause scarring at the base. Same if you masturbate - which has risks no matter what - but at least do not pull it away from your torso against how it naturally goes.
You said you feel something is wrong with it at the base. Trust your body and if you want to make double sure, yes check with a doctor (from my experience they are reluctant to do tests and they rush you, and techs doing the ultrasound will be grabbing your penis and moving it around, without full knowledge or care, but if you'd want to check out a doctor you could) but perhaps what you're feeling down there is pressure? When a penis is so healthy and so erect, the head of the penis can almost be white at parts, because everything is so tight and fully expanded, that yes there is some pressure, everything is rock hard. Feeling that tightness and pressure is a sign of an erection, it's normal.
How I read your post it sounds like you have a healthy erection. That is awesome. Keep it that way. Make sure the girl or you don't pull it against where it naturally goes too much, lay parallel to the girl, as much as you can. If you receive oral sex from the girl (bj) tell her not to pull it down before doing that as much as possible.
I dont think you have a problem when i was in my teens and met a pritty girl i would get an erection my penis would point stright up against my torso and not a problem having sex. Im now 64 my erection stands up at about 20 degrees from my scrotum. dont worry.
At this point, unless the OP comments further, we have no idea if he even masturbates, and if he does, could he have injured his penis that way.
Medically, there is no danger to masturbation, provided it is not being done to the point of skin irritation or chafing, or to the point of excluding other daily activities. As long as any toys or devices are safe, risk of injury is very, very low.
Masturbation doesn't cause scar tissue any more than sex does.
As long as males are not treating their penises like a sticky gear shift in a car, masturbation is fine, and can reduce the risk of prostate cancer.
The best advice for the OP now is to be evaluated by his doctor, preferably a urologist. For now, let's just wait to hear from him, and wish him well.