[RPG] How does the paladin’s Divine Smite damage interact with a Zombie’s Undead Fortitude trait

This is entirely up to the DM.

The rules don't say anywhere whether or not a player automatically knows about whether or not their extra damage triggers on their attacks. The Divine Smite ability in specific doesn't say that, for example, an extra burst of holy light strikes the target, or any other obviously visible display.

Thus, I would say that this is up to your DM and the style of the game that you are playing. If you're playing a game where easily knowing that some of your enemies are undead or fiends would ruin the fun, then you probably do the extra damage invisibly. If you're playing a typical dungeon crawl and it doesn't particularly matter what type of enemy you're fighting, then maybe you could know that the extra damage is being dealt.

It doesn't really matter that much either way. You can always use your Divine Sense to figure out for sure whether they're a fiend or undead, so knowing that you're doing the extra damage wouldn't usually be a big deal.

If the DM decides that this information should be secret, then they have a few options. IF they have a way of secretly rolling the damage die without the player knowing (say, though an automatic die-rolling app, or behind a screen), then they can roll the damage normally. If they can't practically roll the damage without the player knowing, then they can apply the average damage (in this case, 4.5) without telling the player that the extra damage has been added.

Neither affect each other. The cap on extra damage from Divine Smite, as written, applies only to extra damage from Divine Smite. Improved Divine Smite's damage, as written, is in addition to whatever Divine Smite does, and so is not limited by its damage cap.

If the intent was that the Improved Divine Smite damage be considered when calculating the maximum damage doable by Divine Smite, it would say so — saying something like “If you also use your Divine Smite with an attack, this damage counts toward the maximum damage that can be done by Divine Smite.” To the contrary, it gives permission to add this on top of what Divine Smite does.

(The 2018 PHB errata removed the last sentence of Improved Divine Smite, which doesn’t conclusively eliminate this possible confusion, but it does help by no longer being able to read the idea of a shared pool of extra damage out of the text.)