Rema Tip Top are the best of the best in my experience, but most glue on patches are fine. In a pinch, you can use the cement and a piece of old tube to patch a hole, as long as you aren't using an undersized tube in the tire.
The peel-N-stick patches will often get you home, but rarely seal completely and rarely stay on longer than a few days. I kept those on my race bike because they are faster, but they aren't a permanent fix.
As for slime and puncture proof tires, anything that holds air can get a flat. The problem with puncture proof tires is that the thicker and tougher the tire, the less flexible it will be and the more rotating mass you will have. This leads to poorer performance and a harsher ride. So you may have fewer flats, but the time between flats your bike will perform worse and be less pleasant to ride. With Slime or other sealants, you will have more mess to clean up if and when you do hit something that can flatten your tire. However, there is a very narrow range of puncture sizes that slime can't plug, but are still patchable. Generally, if slime can't patch it, you can't either. In those very rare cases where you do actually have a patchable hole that slime failed to patch, you only need to follow a couple extra steps to patch it. Make sure the tube is fully deflated, make sure that the hole is at a higher level than the rest of the tube, and wipe it off with something. A rag, your shirt, a sock, what ever is available. Then patch as normal.
As for slime fouling your valve stem, it takes less time to unscrew the valve stem core, wipe it off, and screw it back in than it likely took you to read this single sentence.
Not that anyone asked, but my personal way of avoiding flats is to run the largest width quality tire my frame and wheel can mount. wider tire means lower ground pressure per square inch, so sharp things have less force to poke through the tire. I run a tube of the closest size to the tire I can. the more a tube stretches, the faster it will deflate after getting a hole. So if I have a bike with a 26"X2.5" tire, and 2 choices of tube, one being 1.75-2.50" and the other being 2.50-3.0", I'd pick the second, as it will be stretching less inside the tire. And I use Slime. In 2 decades, it's never given me an issue, but I have pulled a tube off a bike and counted 110 non leaking punctures after going off trail into a briar patch.