Bike Update
Just wanted to provide an update:
I've successfully completed the replacement of the drivetrain! Everything is working very well. The 9 speed sora shifter works flawlessly with the 9 speed rear MTB derailleur. The Sora front derailleur does shift quite well once set up correctly but it did take a bit of fiddling to get it there (about 20 mins of trial and error). This was mostly an exercise in changing the height of the derailleur and also the cable tension (needs to be quite tight). Another note is that I ended up putting all the bottom bracket spacers on the left (non-drive) side in order to move the chainline over 5mm (reducing it from 50mm to 45mm). I previously had it spaced at 2.5mm to the left (47.5mm chainline) and it wasn't enough.
Here is my final parts list for reference:
Shifters: Shimano Sora ST-R3000(3x9)
Front Derailleur: Shimano Sora FD-R3030
Rear Derailleur: Shimano Alivio RD-M4000
Cassette: Shimano Alivio CS-HG400-9 (11-34)
Crankset: Shimano Deore FC-M590 (44/32/22)
Bottom Bracket: Shimano Deore SM-BB52
This gives me a gear range of 17.6 - 108.7 gear inches. Should be good for getting me up those mountain passes!
Thanks to everyone for their input!
-Alan