Each of the two temporary structures (one in each room) will need to mimic the section of wall that is being removed. So if the jack poles are resting on a single bottom-plate and are holding up a double top-plate, then jack poles 16" on center are called for. If something more substantial than the double top plate and single bottom plate are used, then fewer jack poles are required. In the extreme, it'd take only two jack poles placed 6ft apart to hold up their own temporary 4x10 header. The first floor's floor will need similar support in the basement.
I'd build the temporary support structure using lumber that I could use again somewhere else.
All that is assuming the ceiling joists are perpendicular to the wall in question and that the support structures are placed close to the wall in question.
If instead the ceiling joists are parallel to the wall in question, then there must certainly be a 2nd flr wall that is on top of the wall in question, in which case all the temporary support structures will need to be installed up on the second floor.
Personally, though, my choice would be to do it like this, without shoring up ...
0) pick through the pile at the lumber yard for straight 2x10s
1) open the wall on both sides
2) install the king and jack studs
3) notch the top of the studs on one side to let in one 2x10
4) install the that 2x10 onto the jack studs
5) shim between the cut tops of the studs and 2x10
6) cut off the stud ears (produced by the notching) that are on the side
7) nail off the 9" wide rippings of 1/2" plywood filler
8) install the second 2x10 onto the jack studs
9) nail off the header
10) remove the shims
11) remove the old studs
Standard method is to put a door with a deadbolt on both sides of the doorframe, so both parties have to open their door for the doorway to be open. If not anticipating much use, you might want to pack the space between doors with insulation to reduce sound transmission. And if you don't want them playing with it, no need to give the tenants a key to that particular deadbolt. No need for a knob, either.
Otherwise 2-3 studs, drywall, insulation (again, to block sound as much as possible) and more drywall. Or 4-6 studs (2-3 on each side) that are not connected in the middle for better sound blocking. Double-layer drywall on at least one face might also be worth it for acoustics.
Best Answer
Not if you want it done right, but it's not as bad as you might think. Here's my usual routine to avoid excess drywall repair.
Remove the current door and jamb.
Mark the new rough opening on the wall.
Make the horizontal cuts to widen the rough opening using a circular saw (which will be extremely dusty--have a helper hold a strong vacuum) and/or a reciprocating saw. At this point the current header is still in place, but you've cut through all studs interfering with the new rough opening.
Make the vertical cuts on each side. With luck you'll be between studs. If not, just cut the drywall for now. Remove all drywall inside the R.O. on both sides of the wall.
Remove any studs at the sides of the R.O. that are within 1-1/2" of the R.O. You may need to remove drywall screws and extend the horizontal cut slightly. You should now have voids between drywall panels, at each side of the R.O., of at least 1-1/2" depth into the wall cavity.
At this point you may want to stand your new door in the opening to be sure you have the R.O. correct. You want about 1/2" gap all around.
Remove any drywall screws run into the header. Gently pry the drywall away from the header to locate them. A magnet can help. Once the drywall is completely free, pull down the header. It'll probably be toenailed into the pin studs above and may take substantial force to dislodge.
Once it's out of the way, reach up inside the wall cavity and pry out those nails. You could also cut them off with a rotary tool. The bottoms of the pin studs should be free of obstruction.
If necessary due to a taller replacement header, mark the top of the new header above the rough opening. Make surgical cuts with a reciprocating saw to shorten the pin studs appropriately. If you're very careful, this can be done by cutting through the drywall on one side only.
It's easy to let the saw jerk the studs around and rip up your drywall here. Go easy. You'll have to repair these small slots later anyway, so you can add extra drywall screws to better anchor the pin studs if you like. Remove any drywall screws and the pin stud cutoffs.
Add new king studs where needed. Anchor them in place however you can, by toenailing into the wall plates and with drywall screws. If you have an existing stud less than 6 inches from one side of the R.O. consider extending your header to use that as a king. It's important that both kings be well anchored at the top and bottom, as they're critical to a rigid wall and robust frame for your door. If you can't get nails or screws in place, use a copious amount of construction adhesive (not wood glue).
Slide the new header up into the wall cavity. It might hang there of its own accord, but you could run a couple drywall screws in to temporarily support it, or place a 2x4 leg under it. Toenail or screw the header into the king studs.
Add your trimmer studs. If you used a king that's more than 3" into the wall, add double trimmers on that side, flush with the R.O. You'll want 3" of lumber there for rigidity and for trim backing. Fasten the trimmers to the header and re-fasten the drywall to the trimmers.
Repair the slots and any screw holes in the drywall above the header as necessary.