The projection style you’re going for is usually called “3/4 view” or “RPG view” (because it’s very common in 2D RPGs).
Elevation in this projection is generally purely cosmetic. Cliffs are just walls, and gentle slopes are effectively flat ground. When you walk up a hill or staircase in a game like this, you’re really just walking diagonally on a flat surface. So, you can design your map just for its appearance without worrying about actual height :D
Will these maps need to be loaded into a game engine eventually, or are they just mock-ups? This will change how you should think about game objects (characters, etc).
Tiled only really does one kind of animation: Tile animations, done as frame/flipbook animations. In the Tileset editor, if you select a tile and press the black camera icon, you’ll be taken to the UI to add other tiles as animation frames for this tile. After you’ve done that, any instance of that tile will actually show the animated sequence of the tiles you chose (not all game engines support this feature though!). You can use this to create simple animations in the environment, like lapping waves and spinning.
You can add animated objects like props in a similar way, by creating animated tiles of those things, and placing them as Tile Objects on the map using the Place Tile tool. These will be Objects in that they can be freely moved around without being locked into the grid and can have individual custom properties (useful for telling your game what to do with them), but they’ll also have a tile associated with them, serving as a visual representation of the object. Using Tile Objects is less efficient than regular tiles on Tile Layers, but it’s great for when you need some props or some extra detailing to break the grid (Tile Objects don’t have to be animated!).
More generally, Objects, regular ones and Tile Objects, are usually used to represent anything and everything that isn’t just the world tiles, such as character spawn locations, NPC patrol paths, pick-ups, warp points to other maps, triggers for gameplay and cutscenes, etc. These usually only have meaning when interpreted by your game, so the exact way you’ll use them will depend on your game engine and your specific game code.
Regarding effective use of layers:
Anything goes for mock-ups, but for game use, you generally want to use as few layers as possible. For a 3/4 map, the layer count can usually be kept small: Base tile layer for the terrain (water, grass, cliffs etc) or perhaps a few if the tileset uses transparent transitions instead of dedicated ones, a detail tile layer for static prop tiles like trees and hedges, and some number of Object Groups containing the Objects your gameplay will need. If you’re using Tile Objects for cosmetic props, you’ll want to keep those on their own layer(s) so that you don’t get them mixed up with gameplay stuff.
If your tileset uses transparent transition tiles, you can minimize your total layer count by having terrains that never transition to each other share a layer. For example, if water and cliffs always overlap grass but never each other, then water and cliffs can be on the same layer.
If you haven’t yet, see if your tilesets are compatible with Tiled’s Terrain tool. It helps automate the placing of transitions, saving a lot of time.