Guitar display cabinet

Guitarstorage.com has an excellent product. I keep my 1952 Hauser I in one.

Depends on your humidity level %. Mine are always out, year round never had a single issue.  30 years+

This is a long answer, but it's complicated and you need to determine what's going on and what the source of the moisture is before determining the best solution.

I'm retired but was a professional cabinet maker for 20 years, and then taught wood working for 20 years. I've been building guitars as a hobby for over 50 years so here's my take.

The ideal humidity to aim for is 35 to 45 percent even 30 to 50 should not create any problems. Sudden changes are also more dangerous than gradual change so having the guitar in a climate controlled case where the humidity is maybe 40 then putting it in the car and taking it out in an uncontrolled space where it's 80 percent or vice versa having in a case at 50 then taking it out in a 25 percent environment could cause a problem. In general going from a high humidity to a low humidity is more likely to cause a problem than the other way around, wood can general accommodate some swelling - there might be some action problems but shrinking (going from a high to a low humidty) is what will likely cause cracks. Wood swells up when the humidity goes up and shrinks when it goes down. 

The rusting on your strings can be from 2 sources, the environment or sweaty hands. If you sweat a lot when you play and then put the guitar in the case without wiping off the string and the fingerboard that can cause rusting. The slight chemical content in sweat can actually accelerate the rusting so that is one source. High humidity could be another source, that is water in the air can be condensating on the strings. High humidity alone usually won't do this. It's moisture coming out of the air that can cause problems. An 80 percent humidity means there is a lot of moisture content in the air. Warm air can hold more moisture than cold air. If the air cools then it can't hold as much moisture so some of that moisture comes out of the air. For instance in the summer when you go outside in the morning and there is dew everywhere it's because the warm air has cooled over night and the water has fallen out of the air. Likewise if there is fog in the morning (which is just small water droplets in the air) people talk about the sun "burning" it off. As the temperature rises the air can hold more water so it is dispersed rather than being water. If you are in a hot climate and have air conditioning this can be another source of the water. Warm air from outside is coming into the building and being cooled by the A/C, that means water drops out of the air and lands on everything - it might not be visible but it is still there. In this instance keeping the guitar in the case is better (it slow down the moisture moving to the instrument) it is good to keep a conditioner - planet waves, oasis, boveda and d'addario all have products for this, or you can put a dry sponge (or several) in the case. All of these are meant to suck up some of the moisture keeping the air drier.

This could be happening even inside a case. The case will slow down both the water and temperature changes but it won't stop them completely so if the inside of the case reaches 75F or 20C during the day but the night time temperature drops to 30F or 1C water will condensate - it may not be visible but it would be enough to start rusting the strings.

To start with you should get a couple of hygrometers - you can get them quite inexpensively so get 2 or 3. Put one in the room and one in the case. Check what the readings are so you have an idea of what is going on. The solution might be as simple as putting the guitar out on a stand rather than leave it in the case - the open air of a room allows more circulation and allow moisture to dissipate. Solutions don't need to necessarily be high tech. I was in a guitar museum in Spain that had at least a Million dollars worth of guitars dating from early 1800's through 2000's. They were hung on the wall with glass doors in front but the cases were not sealed. In Spain is very dry so the issue would be keeping the humidity from dropping too low. The solution - plastic drinking cups about half filled with water were placed on a shelf below the instruments! Generally I recommend setting the instrument on a stand in the open air of a room that doesn't get too hot or too cold. An attic room that gets very warm and dry, or a basement that is cool and damp are not good candidates. If your house is climate controlled and you can keep the humidity in the range of 30-50% you should have no trouble just leaving the guitar out.

The classical guitar world is very traditional so your observation (only a little surprised) is right on. The string pressure from a set of nylon strings is half of a set of steel strings, so the neck arguably does not need the rod for strength, but the wood is still subject to moving due to changes in humidity and aging so it's great to have the ability to make a small adjustment. Also because of the lower tension nylon strings are set at a higher action than steel strings so players get used to what I think is a less optimal setting than it could be. The newer truss rods are much lighter so they don't affect the balance the way some of the old ones would.