The wider the footing the less chance of sinking, I always make mine 24 X 24" wide.
One way to figure out pretty close to how much it's going to need to be lifted is with a self leveling lazer level.
It would get set up in the basement.
There so simple to use most people would not even need a manual to use it.
Another simple way is a simple piece of string streched across the whole area inside the house from outside wall to outside wall.
Once it's tight just measure the distance from the string to the floor in the middle of the room. Your also going to need a 6' level to double check inside.
One end gets set in the middle of the room there other end goes toward the wall.
3" is a big lift, things are going to be cracking an popping up stairs.
I hold not lift any more then 1/2" per day. The longer you take to lift it the better. When it's time to lift do not try and lift the whole distance by pumping just one jack. Pump one 2 pumps and move to the next one on down the line, need more then go back and start over. The goal is to try and lift the whole thing as a unit and to avoid humps
Any faster and your going to crack the sheetrock or plaster and may even break some wood.
Plan on having to go back and check all doors and all the walls and doing some min. adjustments and wall repairs.
If I was doing this I would want not less then 4, 20 ton hydrolic jacks.
I would have the jacks sitting on steel 1/2" plate about 12" X 12" on top of the piston I've made up some 4' sq. 1/2 plate with a piece of steel tubing welded together. It gets set on top of the piston, the pipe keeps the plate centered and keeps it from falling off, the plate keep the piston from just crushing a round hole in the wooden 4 X 4 used for lifting.
Once I'm all set up and there's tention againt the beam I insert 3' Long decking screws into the lifting beam through to the main house beam on all 4 sides. This will keep the lifting beam from tipping over as I'm lifting with another jack and taking weight off of it.
Once it's all lifted, if it was ine I would drill clearance through holes for 1/2" carrage bolts every 24", drill one at the top and the next one gets drilled near the bottom, on down the line, This will draw the boards together and make it far more soild. If this beam is going to wraped to hide it at some time then counter bore the holes on both sides so the nut and head of the bolt will be recessed.