Well, you don't have IC, nor should you ever have been suspected of having a UTI. Men hardly ever get either thing, and none of your symptoms should have lead any urologist to think so.
It sounds as though you may have some issue with the bladder neck. I'm surprised that your urologist(s) would have quit so easily. Although they could have been being polite and "indulging" your complaint, figuring you weren't telling them the truth, their overall lackadaisical assessment doesn't seem reasonable.
There could be several things going on, and none of them particularly easy to sort through without a thoughtful and lucky urologist. I say lucky, because some causes can be very difficult to detect.
So, not in any particular order, some of the things that could be going on might be related to a small skin tag in your urethra that gets tears when you strain (like trying to stop flow), weak blood vessel walls (do the sides of your tongue look scalloped when you stick out your tongue?), a parasite, you might be an avid cyclist, a pelvic floor muscle dysfunction, an abnormality in your bladder neck, chronic but mild kidney or prostate stone formation (smaller than sand sized), or chronic anxiety.
Now, there are a lot more questions to ask that might help narrow the field, but those are just a short list of things that could lead to some of the symptoms you described. And then, it might be none of those things.
Yes, straining the muscles in your pelvic region could cause pain and bleeding if you have an abnormality in the lining of your urethra like a skin tag, or some other kind of irritation. That's not conclusive, but just an example.
Urine leakage is usually attributed to a prostate issue or bladder neck issue, however it can also be the result of anxiety.
So, you have a lot of possible sources of your issue, and not enough information to narrow it down very well right now. What do you think might be related?