If you're using the tire for loaded touring, I wouldn't go any smaller than a 38c, a 38 is a good compromise for the road, wide enough so you don't need to put a lot of psi into the tires thus giving you more forgiving ride on the 38, the tire will wear longer than a narrower tire. If you are not doing loaded touring then a 32 will be just fine if it fits the rims.
I don't like slime, it has never worked on high-pressure road tires well.
The first line, and the most important, defense against flats is the tire. On my touring bike I've gone with the Schwalbe Almotion tire, it has the second-best flat protection besides the Marathon Tour Plus tire that is supposed to be flat-proof? really? Anyway, the Almotion tire has the lowest rolling resistance of any touring tire tested, which was why I chose it over the Tour Plus.
The second line of defense against flats would be a tire liner. I only use a tire liner in my touring bike because I don't want to deal with a flat on a loaded bike if I can help it. The tire liner I use is the Clear Motion Rhinodillo because it's much stronger against sharp objects than a Mr Tuffy, I actually tested them both with a tack and I couldn't get the tack to penetrate the Rhino, but I did with the Tuffy, plus the Rhino is lighter in weight and it has a soft edge so as not to chaff the tube and eventually causing a flat like the Tuffy.
The third line of defense is the tube, forget about it, there is no tube made that will prevent a flat, not even a thorn-resistant tube, plus thorn-resistant tubes are very heavy, more than some tires, and they are poorly made. Slime adds weight as well, and it doesn't work that great, there is no need to use any sort of sealant with a great tire and a liner.
I haven't had a flat on my touring bike since I bought the bike in 2019, so the combination of tire and liner is working.