Answer: Yes!  A resounding yes!   Although it might not be immediately obvious, the engine is capable of delivering very complex puzzles, including mechanical manipulation and logic puzzles.  Recall that a "look" region checks whether a given game-state is active, then plays an audio file, then optionally modifies the game-state.  You can use this functionality to have mechanical switches or devices, puzzle elements, game pieces and all sorts of things.  Coupled with the display of a graphical overlay image if the same game-state is set, and you have the makings of rich interactive environments. 

For example, imagine that you want a mechanical lever, which can only be moved if a certain game-state is current.  Draw an overlay image of the lever to suit the current game-state.  Then have a "look" region active at the same time - on the lever.  (Although you should use a special mouse pointer to indicate an "action", rather than a "look".)  When this look region is selected, play a mechanical sound for the streaming audio (and optionally narrate a description of what has just happened).   Have this region set a new game-state.  This new game-state will show the lever in its new position, and can activate new regions, graphics, animations and potential actions anywhere in your game world that you like.

The engine has a simple game-state system, but very complex behaviours can emerge!  Be inventive!

Answer: Since there is no direct way to use an inventory item on another, the solution is to place one inventory item in a scene, and then apply another inventory item to the first one.  The game state can change to reflect the presence of the new "combined object" in the scene, and this new "combined object" can then be picked up.  An alternate approach is to require the items be "combined" before allowing the "combined object" to be picked up (that is, allow the game player to pick up one object, require them to use this on the second item to make a useful "combined object", and then allow them to pick up this "combined object" afterwards).

Again, inventive solutions are possible!

Answer: Create a "once" animation (with a series of overlay images showing the door opening).   When the door is closed, have a region available that allows the user to open the door.  This region can set a "door opening" flag true.  The animation should trigger on this flag.  Once the animation ends, it should in turn set a "door open" flag true, and a new region should allow the user to travel through this door.

Answer: Nothing in a computer is truly random - computer programs simply simulate "randomness" by making it unclear how an event is triggered.  In this way, for all intents and purposes, the event appears to be random to the end user.  In the Adventure Game Engine, one way to achieve randomness is to break up a clickable region into parts.  The coordinates of these parts should combine to appear as the original region, and each part should carry out the same action.   However, each part should set a different game-state flag.  The game player will click on one of these regions, unwittingly setting one of several flags.  When the game needs to trigger a "random event" later on, it can vary the behaviour depending on which of the flags was set earlier.

As far as the game player can tell, it's "random" behaviour!