Is this a load bearing wall

1. Remove trim.
2. Remove door casing.
3. Remove drywall around jack studs (2x4 closest to door opening on each side) and pretty much all the way up to the ceiling. Stop if you have roofing structure above your door - keep going if you don't.
4. Pull out jack studs (you could just extend them with a piece of 2x4 on non load bearing walls)
5. Push up header to desired height. You will have to take out the cripples (small 2x4s) above the header. I don't think the door in your picture is going up a foot with that trim.
6. Put in new longer jack studs on each side.
7. Install new, longer door.
8. Drywall, mud, tape, all that good stuff - this might take the longest.
9. Install new, longer trim. You could use old trim if you could find some casing that will fit on the bottom corners and if door is same width.

It can be kind of hard to tell from photos.

Your house plans (blueprints) would tell you for sure, presuming the house was built faithfully to the plans. You really should have a good look at the plans, or get somebody knowledgeable to look at this in person. Or both.

Do you know which direction the joists are running? Floor and ceiling joists will be perpendicular to the load bearing walls (or beams!), not parallel to them. If the joists pass across the top of the wall (or beam) unbroken, then the wall is possibly not load-bearing, depending on your joists, the length of the span after you remove the wall, and the load on the floor above.

If the ends of the joists are resting on the wall or beam, then it is definitely load-bearing.

if there isn't another floor above this one, and you have engineered, prefabricated trusses holding up your roof, this won't be load-bearing. The trusses are engineered to handle the load of the roof by themselves. If they aren't engineered trusses, that doesn't apply.

Is the long common wall of the living room/dining room/kitchen directly on top of (parallel with) the long wall downstairs, and is the opening between the dining and living rooms perpendicular to that long wall downstairs? It looks like the photos might be from different directions? If so, then the common wall with the kitchen is probably the load-bearing wall, and the opening between the living and dining rooms may or may not be, depending on how the joists are laid out.

But you really should look at the plans or get somebody knowledgeable to take a look in person.