My 2001 subaru legacy gt has a warning light that is flashing rapidly which is supposed to tell me that some door or trunk is open. However, this is doing so even after all doors and trunk are properly closed. What is wrong?

Reset the system by removing both battery terminals
Hold the positive and negative together for 30 seconds
The. Reinstall both terminals
The system will re learn for the next 10 miles or so
If it happens again to the light
We will have to try another approach
Best of luck

There is a diagnostic connector under the dash board that is supposed to be left disconnected (green) if you connect then it forces it into diag mode.....

better to adjust the trunk lock mechanism by trained person.

you gotta check the door switches, one of them is faulty,

Here is a guide for the Subaru Alpine security system:
http://www.cars101.com/subaru/keyless.html#alpine

The link will also list other Subaru systems, but this links to the Alpine portion of the guide. (ie: silence horn, valet mode, etc)

wires must have been trapped,shorted when engine was removed, check loom for damage

the drivers door lock is the main one in the system and its normally the problem when it goes wrong others are effected,you can remove it lubricate the mechanism and try agin or a replacement is needed,

try opening door and holding unlock button ,it sounds like its in valet mode

Legacy, right? Can you tilt the backseats' backrests forward? like when you want to transport something big in the trunk... I suppose the sedan has this similarly to the station wagon. 
Use a pocket torch to look for the lock body from inside and there should be a small button or better a lever that you have no clue what it is there for, right? try to push it or pull it from ech side untill it opens. 
It is actually there to prevent transporting people in the trunk against their will(unlike the movies, where it is possible).

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Yes as long as the Motors are the same. You might need to have the computer out of doner car. Doors are suppose to fit 96 to 99.