Yes, everything you mention can be done in Unity (including profile, inventory and options).
You will have to figure out a way of building levels that will allow you to save them and load them.
You can for instance create a tile based system.
You store info on what item is placed on each tile.
When playing the level you simply instantiate those items at the correct tile.
So basically you need a set of items (door, wall, corner, monster etc) which are placed in a
single dimensional game object array and a 2-dimensional array to represent each tile.
You will need to create an interface for this to work, which might be a bit of a pain, but basically
what you are doing is this:
Tile[0,0] = 1; Where 0,0 is the first tile and 1 is the second object in the game object array.
This can be saved using a variety of methods.
Then when you want to play the level all you need to do is create a loop at level load that goes through the tiles and instantiates each object at it’s correct location.
You can add more information, like rotation or whatever and this can work with lights too (basically any game object).
I think the trickiest part is making a nice interface for the building aspect - as I mentioned above. I guess making a grid and doing a raycast on mouse-clik or touch could get you the tile and if you have some sort of sidebar with the item buttons that simply changes the value to the one used in the game object array you should be able to get something decent running pretty quick.
All items will need to be modular, centered and placement would need to happen centered on the current tile, but this is not that hard to do and you can re-use the code in the level re-construction loop. You will see what I mean once you get the above basics working.
Hope this helps.
K.
PS: This is basically what happens in minecraft, but the array is 3-dimensional, so you can build up and down too.