I put the temp support as close as possible to the exterior wall, since it's needed only for the framing phase, not finishing. You should leave just enough space to slide in the new beam/header over the new window.
Your studs are spaced about 24 o.c. which should be fine, even if the existing wall is studded at 16. After all, this is a temporary wall, and you have control over the loading: get the framing done before snow fall, avoid holiday parties in the attic, and delay any delivery of a grand piano for the upstairs bedroom ;). The point is: temporary support needs to hold a predictable and often lighter load, and only for a short period.
It's also better to preload the new beam, meaning you should lift the top plate of the temporary wall with a jack before removing the old framing, just to the point that you hear the first cracking, and release it once the new header & studs are in place, but before the new window is installed. This ensures that the new header and studs are bearing the load without undue deformation of other structural members, and it also ensures that the the window frame is not bearing load.
The bottom plate of your temp wall should brace several floor joists, or sit on a beam, but it should not run longitudinally over top one joist or, worst of all, between joists on sub-floor. If your bottom plate is perpendicular to the floor joists you are fine. The load on the wall will transfer to several floor joists and via the joists transfer to the foundation.
A diagonal brace for the temp wall is not needed because the existing wall does not appear to be a shear wall. That said, it is all a matter of risk management: if you are not renovating into storm or tornado season, or if feel you can dodge a major earthquake then there is no reason to add the shear brace, but it also does no harm.