Can continuing to pinch my urethra or trying to open it to check for discharge or other abnormalities be causing more harm than good?

If it's not broken..: don't play urologist, Good healthy urination and good healthy ejaculation should put your house in good order, Pinching...not so much.....

Yes: Pinching and trying to open your urethra may cause more harm than good by distracting you from rest of your life. There's absolutely no reason to do so, nor to ruminate about possible diseases. Instead, go see your family doc for evaluation & treatment if necessary. Very unlikely for men to have sexually transmitted infection/disease w/o being aware (more common for women to be asymptomatic).

Not likely going to: Create too much issue, though you could irritate the area. Try to avoid focusing too much on something like this. But if you are worried then just see your doctor for an evaluation. Is there a reason to be worried like a high risk exposure or symptoms?