Yes, the "pee hole" in the penis is the urethra. As Sadie says, the urethra is removed from the penis, cut back to the length necessary, and moved to the place it would be on a cis woman - just above the vagina. (What happens to the removed portion varies; Brassard uses it to form the inside parts of the clitoral hood and inner labia, so that those are pink and mucosal.) Since the internal plumbing isn't affected - the urethra still connects to the bladder and prostate the way it used to - the fluids travel the same way as always. And that does mean ALL of the fluids; it would be complex and difficult (if it's even possible) to reroute ejaculate while leaving urine to emerge from the urethra. In addition, cis women do not ejaculate from the clitoris, so I personally would refuse to go to a surgeon who did something so unusual!
There are no guarantees about what will happen with fluid from the prostate, since some women simply stop producing any due to HRT, surgery, or both. But if you are still capable of producing it after GRS, it will drip out of the urethra and over the vagina. I'm lucky enough that this ends up being a significant help to lubrication, and at this point, I even get wetter when I'm more turned on, just like a cis woman. The only real difference between lubrication in our anatomy and a cis woman's is that some of theirs also "seeps" through the vaginal walls from nearby glands; we have very similar glands, but they work via the urethra as well. However, in my experience, once everything down there is wet and slippery, it's impossible to tell where any of it is coming from anyway.
As an aside, most surgeons don't use the *whole* head of the penis. The glans is much, much larger than the clit on a cis woman, and it would also be distressing for many women to look down there and find a recognizable penis staring back at them. A small portion of the glans (in my case, it's about a quarterr the size of my fingertip) is used to encase the penile nerves, so that the result is all the nerves condensed into a tiny area. This, too, is accurate relative to cis anatomy.
