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I hate to break it to you but a circular saw and Swanson speed square would make cutting the timber for that easier and quicker.
Getting long lengths of construction timber onto a chop saw can be tricky especially as if you don't have a decent stand with long, rigid arms you'll have to spend a while setting something similar up. The other option is to hold down the timber either using a hand (hard and a bit risky) or the clamp (slow)
When I'm doing stud work I follow a process very similar to this:
The main thing I do different is to "crown" the timber (put them in the stack so that bows are all facing the same way).
Also looking at your drawing you need to ensure that there is a stud on both sides of an internal corner (it looks like this isn't the case next to the door). That way you're not going to have the plasterboard flapping around.
Now I've got the saw coming, does this look like a reasonable layout for my studwall? Got to noggin between the joists as one of the floor plates falls between two joists.