What is the best way to stop water from seeping in the door?
 

I have a problem that water is pooling inside the house at the base of the door. As it was raining, I saw that water was running down the brick above the door, around the metal frame of the header and then down the face of the door. Here is a picture of the top of the door.

https://sites.google.com/a/somuchmoo...and%20jamb.jpg

Here is some wood damage at the top of the door. I wonder if that damage was caused by water getting at it from behind the brick and not just running over the surface.

https://sites.google.com/a/somuchmoo...Door%20rot.jpg

Any suggestions on the best way to stop the water?

well
 

It looks like there is no way for the water to get out from behind the brick.

Looks like you have some work to do. The flashing was never installed behind the steel lentil. it should look like the link I have here. Numbers 1 and 2.


http://www.maconline.org/tech/consumers/clinic/leakywindows/leakywindows.html

In the corner of your first photo, we can see what looks like structure above. If this door is covered by a roof or deck, you may have flashing problem up there too.

As Confucius once said (or he should have said, if he actually didn't)--"All of the caulking in the world will not take the place of properly-installed flashing."

So, what is the needed repair
 

If I don't have flashing, does that mean I need to take out a section of brick and install flashing? What is the best course of action?

Is there a roof above this door?
Is there weep holes in the morter above the door?

yes
 

Unfortunatly yes, there is no easy answer. This should have been done when the house was built because it costs so much to fix later on. I would also check any other openings in the brick while your at it.
I know this is not the easy answer, but sadly it is the only one.
If you are not comfortable doing this, then interview and hire a good mason. They will get it the closest to looking right and trying to match mortar.
Good luck.

https://sites.google.com/a/somuchmoo...attredirects=0

There are no weep holes that I can tell. Just above the deck (3-6") is the starting point of the brick and vinyl siding above that.

The proper flashing and weep holes are for water that finds itself on the inside of the brick. If the brick above the door is covered with the deck, they are not the problem. So you want to look at the brick above the deck and the flashing between the siding and the brick. If you can stop the water from getting in there, it would be the cheapist fix, maybe.
Can you post a photo of the deck, brick and siding above?

I see the pics you took of the lintel & door frame header but you were asking about water pooling inside the house @ the bottom of the door...

The pics you show do seem to suggest some sort of water intrusion behind the door frame header which could be caused by the lintel being underflashed or not flashed at all

If the water is running down the outside of the door it could be getting in at the bottom which would suggest an issue with your weatherstrip or door sweep sealing.

Before you start ripping bricks off the house water test the door with a hose - have someone inside watch the bottom corners & the sweep/threshold seal @ the bottom...

If it's the weatherstrip you could purchase a thicker strip & install it (cheap) or buy a storm door

If the problem is at the bottom of the door and you have an adjustable threshold you should also bring it up until you get a good seal & make sure your don't need to replace the sweep (shouldn't be completely flattened or brittle)

As an extra precaution you could also pick up a pack of the weather strip squares (cricket catchers?) that install @ the bottom of the jamb under the weatherstripping to give the water more of a barrier.