I'd say the use of the 1st and 3rd slides can be taught pretty early on. The low D is sharp and needs a bit of slide, and the low C sharp is even more sharp (can be off by 25 cents or more depending on the horn) and needs more slide. These are on the mid-low range of the instrument and are some of the first notes that beginners learn to play, and they sound pretty wildly sharp without a bit of adjustment. Not that you'd need to insist on it to beginners, but it wouldn't hurt to at least mention that the real fingering for a low D is 1+3 + slide.
In my experience when folks say their low Ds and C#s aren't sharp, it's usually because their main tuning slide is too far out and they're playing everything way flat, or the horn just has major tuning issues.
Weather can definitely affect tuning, but it affects it pretty much uniformly. If it's hot out the overall pitch will rise, if it's cold the pitch will drop. You just adjust your main tuning slide accordingly and continue to use your valve slides the same way.
All this is to say I think the use of the slides should be part of the fingering chart. It doesn't take much energy to learn and it makes a huge difference.
Source: I never grew out of being a band kid and play trumpet for a living.
I mean, you can set it up with the handle on the opposite side. Or, my favorite way, you can reach across and drink with it in the crook of your elbow like an old timey moonshine jug.