Trailers are notorious for their lack of corrosion resistance here, a lot of the cheap ones seem to be shot with just paint: no primer or surface prep. They rust big time in Toronto, my dad had a large dump trailer where the floor of the bed started to rust out. He's a contractor and just switch to using bins because it was much easier, and that trailer had all of the electrical problems you would expect. Hydros lasted the entire time with no problems from what I remember, he probably had it for 8-10 years.
My experience with thick, rubberized style bed liners is that they are sticky and hard to sweep clean. You could throw some paint down but you would have to sand and possibly acid prep (if you want the paint to stick) a galvanized surface. I don't think the upside is really worth it. I think I would just get a horse stall mat and cut it to fit, but then you would have to pull it every so often and sweep underneath, but there is still going to be abrasion caused by dust getting under the mat. If you are not hauling large pieces of stone or masonry, you are probably better off to just leave it as is and let it rust. If it is a decent quality trailer stored indoors, it is going to take a LONG time for the corrosion to become an issue. That galvanization is probably more effective than anything you can replace it with.
I think those type of trailers do not hold water like a larger, more enclosed trailer would. Water mixing with debris or leaves really makes them rust. You might want to consider replacing the hardware with stainless though, that is where the corrosion is going to start. Also upgrade the wiring, I would make a new harness with quality components. The main weakness with trailers seems to be their off the shelf components: Axles, bearings, wiring, lights, and fasteners is where the cost cutting is really obvious. You want it ready to rock when you need it, not constantly having to fix nagging issues caused by poor build quality.
Keeping it indoors is going to prolong it's life by a lot. Ata boy!