It can be difficult to find a good repair shop that can speak fluent "Mitsubishi", especially the more rare "Montero" dialect...
The rear differentials usually fair pretty well, unless the companion shaft circlip fails and leaks the gear oil out. Then you can have a rather dramatic failure. Look for evidence on either side of gear oil leaking out. Also, you can remove the fill plug and check the gear oil level. Should be just below the bottom of the hole. Fingertip or allen wrench dipped "over the edge". Circlips are part "N" in the drawing below. Companion Shaft is part "3". The clips install in the groove on the shaft, then slide and "clip" into place in differential housing.
There is an upgraded circlip available from Mitsubishi, and as I understand it, it is a fairly straightforward repair. If there was no high pitched whine at speed, or low roar, I wouldn't be too quick to blame the differential. If you see evidence of gear oil leak from prior, and the fluid level is way low, then you may have an issue. The prior owner of my 2006 Montero was towing a "too heavy" trailer with a JLG scaffold lift and blew out the companion shaft lost all the oil and destroyed the differential.
Here's a video I found showing the issue, but not the repair...
I can't help much with the transmission, except that one time, somehow, my 4WD tried to engage (can't say how or why) at low speed, and I heard a terrible gear grinding noise. Stopped immediately, then checked the gears, turned engine off and on, then presto, back in business. Now I have the 5-speed auto with Selectronic shifter, so I now think one of the sensors/switches was simply sending incorrect gate location info.
It may well be too late for your transmission, but they really favor the genuine Mitsubishi (Hyundai/Kia) SP-III fluid. It is also fairly easy to miss the full mark when checking fluid, especially after a change. Many times folks will fill after a fluid change, and misread as full and hit the road only to discover later they were 2 quarts low, due to pumping through the remote cooler, and not going through the gears, then re-checking the fluid level. Wipe, dip, wipe, dip, because the fluid on the side of the fill tube gives a false reading on the dipstick, until you get it cleaned off enough to get a true reading. Again, I fear you may have bigger issues, but a low fluid level could cause that slip bang issue, especially after sitting and idling a few minutes. When driving around, fluid pressure is up and pumping so it may not have the issue at that time.
The shaking at speed issue, verify you don't have a bad ignition coil causing misfire. That can really shake things up and mimic a transmission issue, although it won't sound like metal grinding. Last, I'd investigate the lift. From your picture, it looks pretty dramatic. I think our Gen 3 independent suspensions don't tolerate too much of a lift. Geometry really takes a hit, especially in the driveshafts. I realize our suspension systems lack the articulation motion of solid axle, but I also think the independent has it's sweet spot in swallowing up the bumpy roads at speed. It's almost brilliant when driven properly. I personally believe the excessive lift takes away those benefits.
What part of the country are you in? Try Car-Part.com for scrap yards with your specific model for a spare transmission. Also tagging my self-appointed driveline "guru"
@Michael Brown, since it's partially his fault I now have two Gen 3 Monteros.
Understand I'm probably too optimistic, but often times what seems to be catastrophe is sometimes misdiagnosed. Best of luck.