The book, at least, is very clear about their route through Moria.
They enter Moria through the West gate, after avoiding the Watcher in the Water and solving the instructions on the door (say "Friend" to enter).
They head up a broad and shallow stairway of 200 steps and through an arch, then pause on the landing at the top to eat.
They set off, and the passage initially twists and turns a little, then heads steadily downwards for a long way, before levelling out.
They continue onward past many openings to other passages, stairways, arches, etc., along with chasms and cracks in the walls and path (at one point they have to leap a 7 foot gap in the ground with churning water a long way below). Gandalf consults often with Gimli about the correct way forwards, but the final choice of path is always left to Gandalf, and they head mostly eastwards as far as they are able.
After several hours since entering Moria they come to an area with a wide arch leading to three eastward heading passages - the left goes down, the middle is level but narrow, and the right heads upwards. Gandalf is unsure which to take. Left of the arch is a stone door leading to a guardroom with an open well in the centre. Pippin throws a stone down the well; after the splash, they hear a few hammer thuds in the deep (possibly signals?) but nothing more comes of it. They rest here. Pippin takes the first watch for one hour, then Gandalf relieves him for six more hours. They decide to take the right hand passage.
They march onward and upward for 8 more hours (with two brief rests). The passage is smooth and climbs steadily in great curves, getting wider and loftier as they go - it was clearly an important thoroughfare. There are no longer any openings at the sides. This passage takes them about 15 miles east, though they walk closer to 20 thanks to the curving.
They come out into a large hall (the Twenty First Hall of the North Side, being on the Seventh Level (with the Eastern entrance to Moria being on the first level)). There is a vast roof high above, held up by many stone pillars. Large arched doorways are in the North, South, East and West sides of the hall (they have entered through the West door, and presume they will need to take the East door). They rest here for the night, with Frodo and then Legolas taking the watch. In the morning they see a shaft above the Eastern arch that leads out to daylight, and also light appearing through the Northern arch.
They explore the Northern arch to find the source of the light. They head down a wide corridor, and see the light is coming from a doorway on the right (i.e. to the east). This room is the Chamber of Mazarbul (the Chamber of Records). It is a large square chamber, dimly lit by a shaft high in the eastern wall heading outside to daylight. Another smaller door lies in the eastern wall under the shaft. Balin's tomb lies in the centre under the light. There are many recesses in the walls containing plundered wooden chests, near which they find a book describing the fate of the dwarven colonists who tried to retake Moria. Bones and weapons lie scattered around both doors where the dwarves made their last stand.
Having learnt from the book where they are (the Twenty First Hall) Gandalf knows they need to head down six levels and to the east and south. He plans to leave the hall by the eastern arch, then head downwards and to the right (i.e. south) when possible.
They are about to head back to the twenty first hall when they hear drums, and many feet heading towards them. Gandalf shines a light in the corridor to see many orcs, some larger black Uruks and at least one giant cave troll. They wedge the western door (that they originally came through) shut, but leave the eastern door open as an escape route. The cave troll forces the door open, getting an arm and foot inside, but retreats when Frodo stabs its foot with Sting. Orcs rush into the room, but retreat after thirteen are slain. The Fellowship make to retreat when a huge orc chieftain leaps into the room. He spears Frodo in the chest, but is killed by Aragorn.
They head through the eastern door. A stairway down is beyond it, which they take. Gandalf remains at the top by the doorway to put a shutting-spell on the door. The Balrog enters the room, and uses a terrible counter-spell which nearly breaks Gandalf. He is forced to use a Word of Command; the door cannot withstand it and breaks apart, then the walls and ceiling of the Chamber of Records collapse, burying Balin's tomb, some of the orcs and the Balrog within, and blocking the way.
They continue down the stairs for an hour, travelling maybe a mile or more. They take no turnings off but continue downwards. All in all they descend seven flights of steps (each maybe 50 or more steps). Finally they come to a low archway that leads through to the Second Hall of Moria, on the First Deep level (i.e. they are now one level below the exit, having headed down seven flights from being six levels above it). The Second Hall is even larger than the Twenty First; two rows of great pillars go down the centre of it. They've come out on the Northern wall of the Hall, nearish to the East side to their left where the exit is, some quarter of a mile away. On their right side, a fissure splits the whole room, with flames leaping out of it. If they had come the usual way into this room they would have been trapped behind the flames and unable to continue.
They head into the hall, turn left and make for the exit. At the eastern end of the Second Hall is one of the defences of old Moria, the Bridge of Khazad Dum. This is a single span of stone with no rails, 50 feet long, wide enough only for people to pass in single file, over a wide chasm in the floor. The orcs now appear behind them, but are themselves trapped behind the fiery fissure they meant to trap the Fellowship with. As the Fellowship reach the bridge, two Cave Trolls appear, placing down giant slabs of stone for the orcs to cross the fissure. The Balrog also reappears. He leaps across the fissure, and races to the bridge.
Gandalf makes a stand in the middle of the bridge, breaking it with his staff before being caught by the Balrog's whip and falling with him. The rest of the bridge breaks.
The Fellowship rush through the eastern door of the hall, and up a broad
stair leading to a wide corridor with shafts of light cut into the roof. Past this is the First Hall of Moria, bright with daylight from windows cut in the east wall. They pass through it to the eastern great gates, which are shattered and broken. Some orcs ambush them, but flee when Aragorn cuts down the leader. Finally they leave Moria through the gateway.
So, to explicitly answer the questions:
Is the Hall of Pillars after they left the stair room the same one or are they two distinct halls?
There are two distinct halls. The Twenty First Hall is the hall they enter before finding Balin's tomb; they then travel for an hour, down seven flights of stairs, before reaching the Second Hall, which contains the bridge.
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) in between them and the bridge?
Yes; a horde of orcs, several cave trolls and the Balrog block the Fellowship's path back to the Twenty First Hall, not to mention tons of rock that fall after Gandalf and the Balrog's spells clash, destroying the tomb.
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 tomb is six levels above the bridge. They head down seven levels, cross the bridge, then need to head one level up again (the broad stairs from the Second Hall to the First Hall) to get to the level of the exit.
What was the route the Balrog took while the Fellowship walked through the stair room?
Presumably, the Balrog dug itself out of the collapsed room and headed back to the Twenty First Hall, then through the Eastern archway. This is the main route down, and the one Gandalf would originally have taken. This path comes out in the Western end of the Second Hall - i.e. behind the fissure with the fire, so it would have trapped the Fellowship if they'd taken their planned route.