OP,
You are merely building a partition - not a supporting wall.
Use 4" x 2" - long lengths are readily available.
Run the floor plate through in a single uncut length - use the loose floor plate to determine where you want the partition to go.
When satisfied Pin the floor plate down - & then drop plumb lines down from the ceiling to the floor plate to find the top plate position on the ceiling.
Fix the top plate (you only need a single top plate), & then fix the bottom plate.
Then fill in with the studs - leave a gap all around the rough opening for the door lining.
Have the door & lining on site - & cut the bottom plate door opening when all dimensions are good.
Use due diligence to find the fixing points in floor & ceiling joists, & avoid utilities.
Consider the door swing, & any elec. switches in the partition.
As you mentioned, Install the p/b's horizontal from the ceiling down. Always cover the opening with a single p/b - dont use cut boards or pieces of board.