You know, it's not actually weird for the desires to be sexual (be it alone or with partners) to wax and wane like you are describing: that's perfectly normal. It's very common for people to have times of high and/or frequent desire, and times of low or infrequent desire, and that can happen day to day, month to month, year to year, decade to decade. It's normal, I promise.
It's not unsafe to touch your urethra, but:
1) You absolutely do not want to try to insert anything in it. It is too small an opening, and it is not anatomically set up for that. That absolutely could do you harm.
2) It is really easy to get a urinary tract infection because of bacteria that gets pushed into the urethra. So, with any genital masturbation, clean hands should be a give, but if you're going to rub that area, it's especially important.
3). There's really no "supposed to" with any kind of pleasure. We are all very diverse in this way, including our bodies. But again, going back to anatomy, the urethra doesn't have the kind of sensory serve endings that say, the glans of the penis or the glans of the clitoris does, nor orifices like the vaginal opening or anus. So, generally, there's not going to be a lot happening there sensation-wise all by itself, but it is surrounded by anatomy where there are more sensory nerve endings. And it's also fine if contact with it feels good.
I'm not sure what you mean about the clitoris not feeling like a penis, but yes, the whole of both systems (most of the clitoris is actually inside the body and not visible) began exactly the same in utero and are highly analagous. Did you have questions about that?