As "a picture is worth a thousand words", especially when talking about movies, I tried to select a few screenshots to highlight the different places.

In their voyage through the Mines of Moria, the Fellowship travels across several great halls with tall columns, and smaller rooms and passages.

They enter the first large hall shortly after Gandalf finally decides which way to choose at the fork he does not remember being in before.

They walk in a straight path along the columns ...

... until Gimli turns right and runs forward entering the Chamber of Mazarbul, where a shaft of light (coming from the opposite side) falls upon the tomb of Balin.

Here they try and barricade against the attack of the goblins and the cave troll.

After defeating the first wave of enemies, they run away from the first column hall (as the goblins are coming from that direction), entering a different large hall.

A large number of goblins quickly runs after them, coming from the room they just left,

and more goblins emerge from wide cracks in the hall floor and ceiling, until they are completely surrounded (note that we cannot see the shaft of light any more, so the heroes have run a long way, or maybe they took a turn into a similar columned passage).

The distant growls of the Balrog scatter the goblins, allowing the Fellowship to run away (note that they seem to be on lower ground than the distant Balrog, so this large hall slopes downward).

The Fellowship run into a side passage too small for the Balrog to enter, so it has to find a different way.

After a dangerous trip down the steep, crumbling flight of stairs

they arrive near the Bridge of Khazad Dum (they arrive running from the right side of the picture; behind Gandalf you can see a large, high-ceilinged ramp they did NOT take).

The Balrog arrives from the ramp and jumps through the flames, while the Fellowship runs towards the bridge, which is now at their level. There are pillars here, too.

After the bridge, they will just have to climb a few steps before finally reaching the Eastern Gate of Moria and exiting.

(note in the background on the right side the remaining parts of the stairs the Fellowship took to arrive here)

Final recap:

Is the Hall of Pillars after they left the stair room the same one or are they two distinct halls?

There are pillars both before and after the steep stairs; they are probably distinct hall connected with large ramps. Note that there is no "stair room", it's more of a chasm crisscrossed by some precariously-looking stairs

If they're one and the same, why did the Fellowship need to go through the room with stairs? Was the Balrog or maybe some other obstacle (there is fire when they leave the stair room) inbetween them and the bridge?

The Balrog had almost reached the Fellowship, so they needed a passage where it could not follow them. Moreover, the large hall had large cracks in the floor that could be impossible to jump over.

How come they needed to go down while the bridge seemed at their level and the Hall of Pillars above them? Did they eventually go any stairs up?

The bridge is at a lower level.

What was the route the Balrog took while the Fellowship walked through the stair room?

He passed through large halls and ramps.